MATH 350 - DISCRETE
MATHEMATICS - Operations on sets
Question Bank - Set 1
Liberty University
Question 1
Question
Let Aand Bbe sets such that |A∩B|= 20, |A\B|= 30, and |B\A|= 40.
Find |A∪B|.
Solution
Step 1: We can use the formula for the cardinality of the union of two sets:
|A∪B|=|A|+|B|−|A∩B|.
Step 2: Given that |A∩B|= 20, |A\B|= 30, and |B\A|= 40, we can
express |A|and |B|in terms of the given values:
|A|=|A\B|+|A∩B|= 30 + 20 = 50
|B|=|B\A|+|A∩B|= 40 + 20 = 60
Step 3: Now substitute the values of |A|and |B|into the formula for |A∪B|:
|A∪B|= 50 + 60 −20 = 90.
Therefore, |A∪B|= 90.
Question 2
Question
Let A={x∈R|x2−3x−10 = 0}and B={x∈R|x2−4x+ 3 = 0}. Find the
intersection A∩Band the union A∪B.
Solution
Step 1: To find the intersection A∩B, we need to find the common elements in
both sets Aand B. Since Aconsists of solutions to the equation x2−3x−10 = 0,
we can factor it as (x−5)(x+ 2) = 0. Thus, A={−2,5}.
Similarly, Bconsists of solutions to the equation x2−4x+ 3 = 0, which
factors as (x−3)(x−1) = 0. Therefore, B={1,3}.
Step 2: Now, let’s find the intersection A∩Bby looking at the common
elements in Aand B. We see that A∩B={1,3}.
Step 3: Next, we will find the union A∪B, which consists of all unique
elements in sets Aand B. Therefore, A∪B={−2,1,3,5}.
Question 3
Question
Let A={x∈Z|2≤x≤10}and B={x∈Z|4≤x≤12}. Find the
intersection and union of sets Aand B.
Solution
Step 1: To find the intersection of sets Aand B, we need to determine the
elements that are common to both sets. Since A={2,3,4,5,6,7,8,9,10}
and B={4,5,6,7,8,9,10,11,12}, the intersection of Aand Bis A∩B=
{4,5,6,7,8,9,10}.
Step 2: To find the union of sets Aand B, we need to combine all elements
from both sets without repetitions. Since A={2,3,4,5,6,7,8,9,10}and B=
{4,5,6,7,8,9,10,11,12}, the union of Aand Bis A∪B={2,3,4,5,6,7,8,9,10,11,12}.
Question 4
Question
Let A={2,4,6,8,10}and B={3,6,9,12}. Define the set C=A∩Band
D=A∪B. Find the cardinality of set Cand set D.
Solution
Step 1: Find C=A∩B, the intersection of sets Aand B.
A∩B={x|x∈Aand x∈B}
={6}
Step 2: Find the cardinality of set C.
|C|=|{6}|
= 1
2
Step 3: Find D=A∪B, the union of sets Aand B.
A∪B={x|x∈Aor x∈B}
={2,3,4,6,8,9,10,12}
Step 4: Find the cardinality of set D.
|D|=|{2,3,4,6,8,9,10,12}|
= 8
Question 5
Question
Let A={x∈N|1≤x≤10}and B={x∈Z| −5≤x≤5}. Find the set
(A∩B)∪(A∪B).
Solution
Step 1: Find A∩B.
A∩B={x∈N|1≤x≤10}∩{x∈Z| −5≤x≤5}
={x∈Z|1≤x≤5}
Step 2: Find A∪B.
A∪B={x∈N|1≤x≤10}∪{x∈Z| −5≤x≤5}
={x∈Z| −5≤x≤10}
Step 3: Find (A∩B)∪(A∪B).
(A∩B)∪(A∪B) = {x∈Z|1≤x≤5}∪{x∈Z| −5≤x≤10}
={x∈Z| −5≤x≤10}
={x∈Z|x∈[−5,10]}
Therefore, (A∩B)∪(A∪B) = {x∈Z| −5≤x≤10}.
Question 6
Question
Let A={x|xis a prime number less than 10}and B={x|xis a positive even number less than 10}.
Determine the sets:
A∪Band A∩B
3
Solution
To find A∪B, we need to find all elements that are in either set Aor set B.
Step 1: Determine the elements of set A:
A={2,3,5,7}
Step 2: Determine the elements of set B:
B={2,4,6,8}
Step 3: Find the union of sets Aand B:
A∪B={2,3,4,5,6,7,8}
Now, to find A∩B, we need to find all elements that are common to both
set Aand set B.
Step 4: Find the intersection of sets Aand B:
A∩B={2}
Therefore, A∪B={2,3,4,5,6,7,8}and A∩B={2}.
Question 7
Question
Let A={x∈Z|3≤x≤10}and B={x∈Z|xis a prime number less than 15}.
Determine A∩B.
Solution
Step 1: Find the elements in set A. Since A={x∈Z|3≤x≤10}, the set A
contains integers from 3 to 10 inclusive. Thus, A={3,4,5,6,7,8,9,10}.
Step 2: Find the elements in set B. The prime numbers less than 15 are 2,
3, 5, 7, 11, and 13. Hence, B={2,3,5,7,11,13}.
Step 3: Determine A∩B. The intersection of sets Aand Bconsists of the
elements that are common in both Aand B. Therefore, A∩B={3,5,7}.
Question 8
Question
Let A,B, and Cbe sets such that A={x∈Z|1≤x≤10},B={x∈Z|5≤
x≤15}, and C={x∈Z|8≤x≤12}. Find (A∪B)∩C.
4
Solution
Step 1: We first find A∪B, the union of sets Aand B.
Let’s list the elements of set A:A={1,2,3,4,5,6,7,8,9,10}
Let’s list the elements of set B:B={5,6,7,8,9,10,11,12,13,14,15}
A∪B={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
Step 2: Next, we find (A∪B)∩C, the intersection of sets A∪Band C.
Let’s list the elements of set C:C={8,9,10,11,12}
(A∪B)∩C={8,9,10,11,12}
Thus, (A∪B)∩C={8,9,10,11,12}.
Question 9
Question
Let A={x∈N:xis a prime number less than 10}and B={1,2,3,4,5}.
Find A∩B.
Solution
To find the intersection A∩B, we need to determine the set of elements that
are common to both Aand B.
Step 1: Determine the prime numbers less than 10. The prime numbers
less than 10 are: 2, 3, 5, 7. Therefore, A={2,3,5,7}.
Step 2: Determine the intersection of Aand B. The intersection of Aand
Bis the set of elements that are common to both sets. In this case, A∩B=
{2,3,5}.
Therefore, A∩B={2,3,5}.
Question 10
Question
Let A={x∈Z|2≤x≤8}and B={x∈Z|4≤x≤10}. Find A∩Band
A∪B.
5
Solution
To find A∩B(the intersection of Aand B), weneedtodeterminetheelementsthatarecommontobothsetsAandB.
Step 1: List out the elements in sets Aand B.
A={2,3,4,5,6,7,8}
B={4,5,6,7,8,9,10}
Step 2: Identify the elements that are common to both sets.
A∩B={4,5,6,7,8}
Now, let’s find A∪B(the union of Aand B), whichcontainsalldistinctelementsfrombothsetsAandB.
Step 3: List out the elements in sets Aand B.
A={2,3,4,5,6,7,8}
B={4,5,6,7,8,9,10}
Step 4: Merge the elements from sets Aand B, removingduplicates.A∪B=
{2,3,4,5,6,7,8,9,10}
Therefore, A∩B={4,5,6,7,8}and A∪B={2,3,4,5,6,7,8,9,10}.
Question 11
Question
Let A={x∈Z|2≤x≤10}and B={x∈Z|5≤x≤15}. Find
(A∩B)∪(A\B).
Solution
Step 1: We first find A∩B, which represents the elements that are in both sets
Aand B.
A∩B={x∈Z|5≤x≤10}
Step 2: Next, we find A\B, which represents the elements that are in set
Abut not in set B.
A\B={x∈Z|2≤x < 5}
Step 3: Now, we find (A∩B)∪(A\B), which is the union of the elements
in A∩Band A\B.
(A∩B)∪(A\B) = {x∈Z|2≤x≤10 and x= 11,12,13,14,15}
Step 4: Simplifying the set obtained in the previous step, we have
(A∩B)∪(A\B) = {2,3,4,5,6,7,8,9,10}
Therefore, (A∩B)∪(A\B) = {2,3,4,5,6,7,8,9,10}.
6
Question 12
Question
Let Aand Bbe sets such that |A∪B|= 30, |A∩B|= 10, and |A−B|= 20.
Find |B|.
Solution
Step 1: We know that the number of elements in the union of two sets Aand
Bis given by the formula:
|A∪B|=|A|+|B|−|A∩B|.
Substitute the given values into this formula:
30 = |A|+|B| − 10.
Step 2: We also know that the number of elements in the set difference A−B
is given by the formula:
|A−B|=|A|−|A∩B|.
Substitute the given values into this formula:
20 = |A| − 10.
Step 3: Solve the system of equations from Step 1 and Step 2 to find |A|
and |B|. From Step 2, we have |A|= 30, and substituting this into the equation
from Step 1 gives:
30 = 30 + |B| − 10.
Step 4: Simplify the equation to solve for |B|:
|B|= 10.
Step 5: Therefore, the number of elements in set B, denoted by |B|, is 10.
Question 13
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|5≤x≤15}. Find the set
A∩B.
7
Solution
Step 1: To find the intersection A∩B, we need to determine the elements that
are common to both sets Aand B.
Step 2: Set Acontains the integers from 1 to 10, while set Bcontains the
integers from 5 to 15.
Step 3: The elements that are common to both sets are the integers that
satisfy both conditions. These integers are in the range from 5 to 10 since they
are present in both sets.
Step 4: Therefore, A∩B={x∈Z|5≤x≤10}.
Question 14
Question
Let A={x∈R|0< x ≤5}and B={x∈R|3<x<8}. Find the set A∩B.
Solution
Step 1: To find A∩B, we need to identify the elements that are common to
both sets Aand B.
Step 2: Set Aconsists of all real numbers greater than 0 and less than or
equal to 5, while set Bconsists of all real numbers greater than 3 and less than
8.
Step 3: We need to determine the intersection of sets Aand B, which includes
real numbers that satisfy both conditions (greater than 0 and less than or equal
to 5, and greater than 3 and less than 8).
Step 4: The intersection of sets Aand Bcan be defined as A∩B={x∈
R|3< x ≤5}.
Therefore, A∩B={x∈R|3< x ≤5}.
Question 15
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|5≤x≤15}. Determine the
following sets:
1. A∪B
2. A∩B
3. A\B
4. B\A
8
Solution
Given: A={x∈Z|1≤x≤10}B={x∈Z|5≤x≤15}
1. To find A∪B, we need to list all unique elements in Aand Bcombined:
A∪B={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
2. To find A∩B, we need to find the elements that are common in both A
and B:
A∩B={5,6,7,8,9,10}
3. To find A\B, we need to list the elements in Athat are not in B:
A\B={1,2,3,4}
4. To find B\A, we need to list the elements in Bthat are not in A:
B\A={11,12,13,14,15}
Question 16
Question
Let A={x∈Z|3≤x≤10}and B={x∈Z|5≤x≤12}. Find the set
A∩Band write the elements in ascending order.
Solution
Step 1: List the elements of Aand B.
A={3,4,5,6,7,8,9,10}
B={5,6,7,8,9,10,11,12}
Step 2: Find the intersection of sets Aand B.
A∩B={x|x∈Aand x∈B}
Step 3: Determine the common elements in both sets Aand B.
A∩B={5,6,7,8,9,10}
Step 4: Arrange the elements in ascending order.
A∩B={5,6,7,8,9,10}
Therefore, A∩B={5,6,7,8,9,10}.
9
Question 17
Question
Let A={x∈Z|0≤x≤10}and B={x∈Z|5≤x≤15}. Determine the
set A∩B.
Solution
Step 1: To find A∩B, we need to identify the elements that are present in both
sets Aand B.
Step 2: Set Acontains the integers from 0 to 10, while set Bcontains the
integers from 5 to 15.
Step 3: The intersection of Aand Bwill include the integers that are common
to both sets, namely the integers from 5 to 10 (inclusive).
Step 4: Therefore, the set A∩Bis {5,6,7,8,9,10}.
Question 18
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|xis prime}. Find the set
(A∪B)∩(A∩B).
Solution
Step 1: Let’s first determine the elements in set A:
A={1,2,3,4,5,6,7,8,9,10}
Step 2: Next, we determine the elements in set B:
B={2,3,5,7}
Step 3: Now, we find the union of sets Aand B:
A∪B={1,2,3,4,5,6,7,8,9,10}∪{2,3,5,7}={1,2,3,4,5,6,7,8,9,10}
Step 4: Next, we find the intersection of sets Aand B:
A∩B={1,2,3,4,5,6,7,8,9,10}∩{2,3,5,7}={2,3,5,7}
Step 5: Finally, we find the intersection of (A∪B) and (A∩B):
(A∪B)∩(A∩B) = {1,2,3,4,5,6,7,8,9,10}∩{2,3,5,7}={2,3,5,7}
Therefore, the set (A∪B)∩(A∩B) is {2,3,5,7}.
10
Question 19
Question
Let A={x∈Z|1≤x≤10},B={x∈Z|5≤x≤15}, and C={x∈Z|
xis prime}. Find (A∪B)−C.
Solution
Step 1: Find A∪B
A∪B={x∈Z|1≤x≤10}∪{x∈Z|5≤x≤15}
={x∈Z|1≤x≤15}
Step 2: Find (A∪B)−C
(A∪B)−C={x∈Z|1≤x≤15}−{x∈Z|xis prime}
={2,4,6,8,10,12,14}
Therefore, (A∪B)−C={2,4,6,8,10,12,14}.
Question 20
Question
Let Aand Bbe sets such that A={x∈Z|2≤x≤10}and B={x∈Z|5≤
x≤15}. Find the intersection and union of sets Aand B.
Solution
Step 1: Find the intersection of sets Aand B.
A∩B={x∈Z|x∈Aand x∈B}
={x∈Z|2≤x≤10 and 5 ≤x≤15}
={x∈Z|5≤x≤10}
={5,6,7,8,9,10}
Step 2: Find the union of sets Aand B.
A∪B={x∈Z|x∈Aor x∈B}
={x∈Z|2≤x≤10 or 5 ≤x≤15}
={x∈Z|2≤x≤15}
={2,3,4,5,6,7,8,9,10,11,12,13,14,15}
11
Question 21
Question
Let A,B, and Cbe sets such that |A∩B|= 12, |B∩C|= 15, |A∩C|= 20,
|A∩B∩C|= 7, |A∪B∪C|= 50, and |A∩B∩C′|= 5. Determine the number
of elements in |A∪B∪C′|.
Solution
Step 1: Use the principle of inclusion-exclusion to find the number of elements
in the union of the three sets A,B, and C.
|A∪B∪C|=|A|+|B|+|C|−|A∩B|−|B∩C|−|A∩C|+|A∩B∩C|
=|A|+|B|+|C| − 12 −15 −20 + 7
=|A|+|B|+|C| − 40
Step 2: Substituting the given information, we have
50 = |A|+|B|+|C| − 40
|A|+|B|+|C|= 90
Step 3: Use the fact |A∩B′∩C|=|A|−|A∩B|=|A| − 12 to determine
|A∩B′∩C|.
|A∩B′∩C|=|A|−|A∩B|
=|A| − 12
Step 4: Apply the principle of inclusion-exclusion to find the number of
elements in A∪B∪C′.
|A∪B∪C′|=|A|+|B|+|C|−|A∩B|−|B∩C|−|A∩C|+|A∩B∩C|+|A∩B′∩C|
= 90 −12 −15 −20 + 7 + (|A| − 12)
= 90 −40 −15 −20 −12 + 7
= 10
Therefore, the number of elements in |A∪B∪C′|is 10 .
Question 22
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|xis prime}. Determine the set
(A∩B)∪(A∩Bc).
12
Solution
Step 1: Find the intersection of set Aand set B.
A={x∈Z|1≤x≤10}
={1,2,3,4,5,6,7,8,9,10}
B={x∈Z|xis prime}
={2,3,5,7}
A∩B={2,3,5,7}
Step 2: Find the complement of set B. The complement of Bdenoted by Bc
is all integers between 1 and 10 that are not prime. This set is {1,4,6,8,9,10}.
Step 3: Find the intersection of set Aand set Bc.
A∩Bc={1,4,6,8,9,10}
Step 4: Combine the results from Step 1 and Step 3.
(A∩B)∪(A∩Bc) = {2,3,5,7}∪{1,4,6,8,9,10}
={1,2,3,4,5,6,7,8,9,10}
Therefore, (A∩B)∪(A∩Bc) = {1,2,3,4,5,6,7,8,9,10}.
Question 23
Question
Let A={x∈N|3≤x≤9}and B={x∈N|4≤x≤10}. Find A∩Band
A∪B.
Solution
Let’s first determine the elements in the sets Aand Bbefore finding their
intersection and union.
Step 1: Determine the elements in set ASet Acontains natural num-
bers xsuch that 3 ≤x≤9. Therefore, A={3,4,5,6,7,8,9}.
Step 2: Determine the elements in set BSet Bcontains natural num-
bers xsuch that 4 ≤x≤10. Therefore, B={4,5,6,7,8,9,10}.
Step 3: Find the intersection of sets Aand BThe intersection of
sets Aand B, denoted A∩B, contains elements that are present in both sets.
Therefore, A∩B={4,5,6,7,8,9}.
Step 4: Find the union of sets Aand BThe union of sets Aand B,
denoted A∪B, contains all elements that are present in either set. Therefore,
A∪B={3,4,5,6,7,8,9,10}.
13
Question 24
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|5≤x≤15}. Find
(A∪B)∩(AC∪BC).
Solution
Step 1: We first find A∪B.
A∪B={x∈Z|1≤x≤10}∪{x∈Z|5≤x≤15}
={x∈Z|1≤x≤15}
={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
Step 2: Next, we find ACand BC.
AC={x∈Z|x /∈A}={x∈Z|x≤0 or x > 10}
={x∈Z|x≤0}
={. . . , −3,−2,−1,0}
BC={x∈Z|x /∈B}={x∈Z|x < 5 or x > 15}
={x∈Z|x < 5}∪{x∈Z|x > 15}
={. . . , −3,−2,−1,0,1,2,3,4}∪{16,17,18,19, . . . }
Step 3: Now, we find (A∪B)∩(AC∪BC).
(A∪B)∩(AC∪BC) = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}∩
{. . . , −3,−2,−1,0,1,2,3,4}∪{16,17,18,19, . . . }
={1,2,3,4,5}
Therefore, (A∪B)∩(AC∪BC) = {1,2,3,4,5}.
Question 25
Question
Let A={x∈R|x2−3x+ 2 <0}and B={x∈R|2x−5<3}. Find A∩B.
Solution
Step 1: First, let’s find the values of xthat satisfy the inequality x2−3x+2 <0.
x2−3x+ 2 <0
(x−1)(x−2) <0
14
The solutions to the inequality x2−3x+ 2 <0 are 1 <x<2.
Step 2: Next, let’s find the values of xthat satisfy the inequality 2x−5<3.
2x−5<3
2x < 8
x < 4
The solutions to the inequality 2x−5<3 are x < 4.
Step 3: To find A∩B, we need to find the values of xthat satisfy both
1<x<2 and x < 4. The solutions that satisfy both inequalities are 1 <x<2.
Therefore, A∩B={x∈R|1<x<2}.
Question 26
Question
Let A={x∈Z|2< x ≤10}and B={x∈N|5< x ≤15}. Find the set
A∩B.
Solution
Step 1: Since A={x∈Z|2< x ≤10}, we have A={3,4,5,6,7,8,9,10}.
Step 2: Since B={x∈N|5< x ≤15}, we have B={6,7,8,9,10,11,12,13,14,15}.
Step 3: To find A∩B, we need to find the elements common to both sets A
and B.
Step 4: Therefore, A∩B={6,7,8,9,10}.
Hence, A∩B={6,7,8,9,10}.
Question 27
Question
Let A,B, and Cbe sets such that A∪B=A∪Cand A∩B=A∩C. Prove
that B=C.
Solution
We will prove that B=Cby showing that B⊆Cand C⊆B.
Step 1: Show that B⊆C
Assume x∈B. Since A∩B=A∩C, we have x∈Aif and only if x∈C.
Thus, x∈Bimplies x∈C, which shows that B⊆C.
Step 2: Show that C⊆B
Assume y∈C. Since A∪B=A∪C, we have y∈Aor y∈Bif and only
if y∈Aor y∈C. Since y∈A(from the fact that A∩B=A∩C), we can
conclude that y∈B, which shows that C⊆B.
Since we have shown that B⊆Cand C⊆B, we can conclude that B=C.
15
Question 28
Question
Let A,B, and Cbe sets such that A∪B=A∪Cand A∩B=A∩C. Prove
that B=C.
Solution
Step 1: Let’s start by proving that B⊆C. Since A∩B=A∩C, we have
A∩B⊆A∩Cby definition of subset. Then, for any x∈B, we have x∈A∩B,
which implies x∈A∩C. Thus, B⊆C.
Step 2: Next, we will prove that C⊆B. Since A∩B=A∩C, we have
A∩C⊆A∩Bby definition of subset. Then, for any y∈C, we have y∈A∩C,
which implies y∈A∩B. Thus, C⊆B.
Step 3: Combining Step 1 and Step 2, we conclude that B=Csince B⊆C
and C⊆B.
Question 29
Question
Let A={x∈Z: 2 ≤x≤8}and B={x∈Z: 5 ≤x≤12}. Find the set
A∪B.
Solution
Step 1: List out the elements of sets Aand Bindividually.
A={2,3,4,5,6,7,8}
B={5,6,7,8,9,10,11,12}
Step 2: Combine the elements of Aand Bto form the union A∪B.
,6,7,8, 9, 10, 11, 12}
={2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Therefore, A∪B={2,3,4,5,6,7,8,9,10,11,12}.
Question 30
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|6≤x≤15}. Find the set
A∪B.
16
Solution
Step 1: To find the intersection A∩B, we need to find the common elements in
both sets Aand B. Since Aconsists of solutions to the equation x2−3x−10 = 0,
we can factor it as (x−5)(x+ 2) = 0. Thus, A={−2,5}.
Similarly, Bconsists of solutions to the equation x2−4x+ 3 = 0, which
factors as (x−3)(x−1) = 0. Therefore, B={1,3}.
Step 2: Now, let’s find the intersection A∩Bby looking at the common
elements in Aand B. We see that A∩B={1,3}.
Step 3: Next, we will find the union A∪B, which consists of all unique
elements in sets Aand B. Therefore, A∪B={−2,1,3,5}.
Question 3
Question
Let A={x∈Z|2≤x≤10}and B={x∈Z|4≤x≤12}. Find the
intersection and union of sets Aand B.
Solution
Step 1: To find the intersection of sets Aand B, we need to determine the
elements that are common to both sets. Since A={2,3,4,5,6,7,8,9,10}
and B={4,5,6,7,8,9,10,11,12}, the intersection of Aand Bis A∩B=
{4,5,6,7,8,9,10}.
Step 2: To find the union of sets Aand B, we need to combine all elements
from both sets without repetitions. Since A={2,3,4,5,6,7,8,9,10}and B=
{4,5,6,7,8,9,10,11,12}, the union of Aand Bis A∪B={2,3,4,5,6,7,8,9,10,11,12}.
Question 4
Question
Let A={2,4,6,8,10}and B={3,6,9,12}. Define the set C=A∩Band
D=A∪B. Find the cardinality of set Cand set D.
Solution
Step 1: Find C=A∩B, the intersection of sets Aand B.
A∩B={x|x∈Aand x∈B}
={6}
Step 2: Find the cardinality of set C.
|C|=|{6}|
= 1
2
Step 3: Find D=A∪B, the union of sets Aand B.
A∪B={x|x∈Aor x∈B}
={2,3,4,6,8,9,10,12}
Step 4: Find the cardinality of set D.
|D|=|{2,3,4,6,8,9,10,12}|
= 8
Question 5
Question
Let A={x∈N|1≤x≤10}and B={x∈Z| −5≤x≤5}. Find the set
(A∩B)∪(A∪B).
Solution
Step 1: Find A∩B.
A∩B={x∈N|1≤x≤10}∩{x∈Z| −5≤x≤5}
={x∈Z|1≤x≤5}
Step 2: Find A∪B.
A∪B={x∈N|1≤x≤10}∪{x∈Z| −5≤x≤5}
={x∈Z| −5≤x≤10}
Step 3: Find (A∩B)∪(A∪B).
(A∩B)∪(A∪B) = {x∈Z|1≤x≤5}∪{x∈Z| −5≤x≤10}
={x∈Z| −5≤x≤10}
={x∈Z|x∈[−5,10]}
Therefore, (A∩B)∪(A∪B) = {x∈Z| −5≤x≤10}.
Question 6
Question
Let A={x|xis a prime number less than 10}and B={x|xis a positive even number less than 10}.
Determine the sets:
A∪Band A∩B
3
Solution
To find A∪B, we need to find all elements that are in either set Aor set B.
Step 1: Determine the elements of set A:
A={2,3,5,7}
Step 2: Determine the elements of set B:
B={2,4,6,8}
Step 3: Find the union of sets Aand B:
A∪B={2,3,4,5,6,7,8}
Now, to find A∩B, we need to find all elements that are common to both
set Aand set B.
Step 4: Find the intersection of sets Aand B:
A∩B={2}
Therefore, A∪B={2,3,4,5,6,7,8}and A∩B={2}.
Question 7
Question
Let A={x∈Z|3≤x≤10}and B={x∈Z|xis a prime number less than 15}.
Determine A∩B.
Solution
Step 1: Find the elements in set A. Since A={x∈Z|3≤x≤10}, the set A
contains integers from 3 to 10 inclusive. Thus, A={3,4,5,6,7,8,9,10}.
Step 2: Find the elements in set B. The prime numbers less than 15 are 2,
3, 5, 7, 11, and 13. Hence, B={2,3,5,7,11,13}.
Step 3: Determine A∩B. The intersection of sets Aand Bconsists of the
elements that are common in both Aand B. Therefore, A∩B={3,5,7}.
Question 8
Question
Let A,B, and Cbe sets such that A={x∈Z|1≤x≤10},B={x∈Z|5≤
x≤15}, and C={x∈Z|8≤x≤12}. Find (A∪B)∩C.
4
Solution
Step 1: We first find A∪B, the union of sets Aand B.
Let’s list the elements of set A:A={1,2,3,4,5,6,7,8,9,10}
Let’s list the elements of set B:B={5,6,7,8,9,10,11,12,13,14,15}
A∪B={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
Step 2: Next, we find (A∪B)∩C, the intersection of sets A∪Band C.
Let’s list the elements of set C:C={8,9,10,11,12}
(A∪B)∩C={8,9,10,11,12}
Thus, (A∪B)∩C={8,9,10,11,12}.
Question 9
Question
Let A={x∈N:xis a prime number less than 10}and B={1,2,3,4,5}.
Find A∩B.
Solution
To find the intersection A∩B, we need to determine the set of elements that
are common to both Aand B.
Step 1: Determine the prime numbers less than 10. The prime numbers
less than 10 are: 2, 3, 5, 7. Therefore, A={2,3,5,7}.
Step 2: Determine the intersection of Aand B. The intersection of Aand
Bis the set of elements that are common to both sets. In this case, A∩B=
{2,3,5}.
Therefore, A∩B={2,3,5}.
Question 10
Question
Let A={x∈Z|2≤x≤8}and B={x∈Z|4≤x≤10}. Find A∩Band
A∪B.
5
Solution
To find A∩B(the intersection of Aand B), weneedtodeterminetheelementsthatarecommontobothsetsAandB.
Step 1: List out the elements in sets Aand B.
A={2,3,4,5,6,7,8}
B={4,5,6,7,8,9,10}
Step 2: Identify the elements that are common to both sets.
A∩B={4,5,6,7,8}
Now, let’s find A∪B(the union of Aand B), whichcontainsalldistinctelementsfrombothsetsAandB.
Step 3: List out the elements in sets Aand B.
A={2,3,4,5,6,7,8}
B={4,5,6,7,8,9,10}
Step 4: Merge the elements from sets Aand B, removingduplicates.A∪B=
{2,3,4,5,6,7,8,9,10}
Therefore, A∩B={4,5,6,7,8}and A∪B={2,3,4,5,6,7,8,9,10}.
Question 11
Question
Let A={x∈Z|2≤x≤10}and B={x∈Z|5≤x≤15}. Find
(A∩B)∪(A\B).
Solution
Step 1: We first find A∩B, which represents the elements that are in both sets
Aand B.
A∩B={x∈Z|5≤x≤10}
Step 2: Next, we find A\B, which represents the elements that are in set
Abut not in set B.
A\B={x∈Z|2≤x < 5}
Step 3: Now, we find (A∩B)∪(A\B), which is the union of the elements
in A∩Band A\B.
(A∩B)∪(A\B) = {x∈Z|2≤x≤10 and x= 11,12,13,14,15}
Step 4: Simplifying the set obtained in the previous step, we have
(A∩B)∪(A\B) = {2,3,4,5,6,7,8,9,10}
Therefore, (A∩B)∪(A\B) = {2,3,4,5,6,7,8,9,10}.
6
Question 12
Question
Let Aand Bbe sets such that |A∪B|= 30, |A∩B|= 10, and |A−B|= 20.
Find |B|.
Solution
Step 1: We know that the number of elements in the union of two sets Aand
Bis given by the formula:
|A∪B|=|A|+|B|−|A∩B|.
Substitute the given values into this formula:
30 = |A|+|B| − 10.
Step 2: We also know that the number of elements in the set difference A−B
is given by the formula:
|A−B|=|A|−|A∩B|.
Substitute the given values into this formula:
20 = |A| − 10.
Step 3: Solve the system of equations from Step 1 and Step 2 to find |A|
and |B|. From Step 2, we have |A|= 30, and substituting this into the equation
from Step 1 gives:
30 = 30 + |B| − 10.
Step 4: Simplify the equation to solve for |B|:
|B|= 10.
Step 5: Therefore, the number of elements in set B, denoted by |B|, is 10.
Question 13
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|5≤x≤15}. Find the set
A∩B.
7
Solution
Step 1: To find the intersection A∩B, we need to determine the elements that
are common to both sets Aand B.
Step 2: Set Acontains the integers from 1 to 10, while set Bcontains the
integers from 5 to 15.
Step 3: The elements that are common to both sets are the integers that
satisfy both conditions. These integers are in the range from 5 to 10 since they
are present in both sets.
Step 4: Therefore, A∩B={x∈Z|5≤x≤10}.
Question 14
Question
Let A={x∈R|0< x ≤5}and B={x∈R|3<x<8}. Find the set A∩B.
Solution
Step 1: To find A∩B, we need to identify the elements that are common to
both sets Aand B.
Step 2: Set Aconsists of all real numbers greater than 0 and less than or
equal to 5, while set Bconsists of all real numbers greater than 3 and less than
8.
Step 3: We need to determine the intersection of sets Aand B, which includes
real numbers that satisfy both conditions (greater than 0 and less than or equal
to 5, and greater than 3 and less than 8).
Step 4: The intersection of sets Aand Bcan be defined as A∩B={x∈
R|3< x ≤5}.
Therefore, A∩B={x∈R|3< x ≤5}.
Question 15
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|5≤x≤15}. Determine the
following sets:
1. A∪B
2. A∩B
3. A\B
4. B\A
8
Solution
Given: A={x∈Z|1≤x≤10}B={x∈Z|5≤x≤15}
1. To find A∪B, we need to list all unique elements in Aand Bcombined:
A∪B={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
2. To find A∩B, we need to find the elements that are common in both A
and B:
A∩B={5,6,7,8,9,10}
3. To find A\B, we need to list the elements in Athat are not in B:
A\B={1,2,3,4}
4. To find B\A, we need to list the elements in Bthat are not in A:
B\A={11,12,13,14,15}
Question 16
Question
Let A={x∈Z|3≤x≤10}and B={x∈Z|5≤x≤12}. Find the set
A∩Band write the elements in ascending order.
Solution
Step 1: List the elements of Aand B.
A={3,4,5,6,7,8,9,10}
B={5,6,7,8,9,10,11,12}
Step 2: Find the intersection of sets Aand B.
A∩B={x|x∈Aand x∈B}
Step 3: Determine the common elements in both sets Aand B.
A∩B={5,6,7,8,9,10}
Step 4: Arrange the elements in ascending order.
A∩B={5,6,7,8,9,10}
Therefore, A∩B={5,6,7,8,9,10}.
9
Question 17
Question
Let A={x∈Z|0≤x≤10}and B={x∈Z|5≤x≤15}. Determine the
set A∩B.
Solution
Step 1: To find A∩B, we need to identify the elements that are present in both
sets Aand B.
Step 2: Set Acontains the integers from 0 to 10, while set Bcontains the
integers from 5 to 15.
Step 3: The intersection of Aand Bwill include the integers that are common
to both sets, namely the integers from 5 to 10 (inclusive).
Step 4: Therefore, the set A∩Bis {5,6,7,8,9,10}.
Question 18
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|xis prime}. Find the set
(A∪B)∩(A∩B).
Solution
Step 1: Let’s first determine the elements in set A:
A={1,2,3,4,5,6,7,8,9,10}
Step 2: Next, we determine the elements in set B:
B={2,3,5,7}
Step 3: Now, we find the union of sets Aand B:
A∪B={1,2,3,4,5,6,7,8,9,10}∪{2,3,5,7}={1,2,3,4,5,6,7,8,9,10}
Step 4: Next, we find the intersection of sets Aand B:
A∩B={1,2,3,4,5,6,7,8,9,10}∩{2,3,5,7}={2,3,5,7}
Step 5: Finally, we find the intersection of (A∪B) and (A∩B):
(A∪B)∩(A∩B) = {1,2,3,4,5,6,7,8,9,10}∩{2,3,5,7}={2,3,5,7}
Therefore, the set (A∪B)∩(A∩B) is {2,3,5,7}.
10
Question 19
Question
Let A={x∈Z|1≤x≤10},B={x∈Z|5≤x≤15}, and C={x∈Z|
xis prime}. Find (A∪B)−C.
Solution
Step 1: Find A∪B
A∪B={x∈Z|1≤x≤10}∪{x∈Z|5≤x≤15}
={x∈Z|1≤x≤15}
Step 2: Find (A∪B)−C
(A∪B)−C={x∈Z|1≤x≤15}−{x∈Z|xis prime}
={2,4,6,8,10,12,14}
Therefore, (A∪B)−C={2,4,6,8,10,12,14}.
Question 20
Question
Let Aand Bbe sets such that A={x∈Z|2≤x≤10}and B={x∈Z|5≤
x≤15}. Find the intersection and union of sets Aand B.
Solution
Step 1: Find the intersection of sets Aand B.
A∩B={x∈Z|x∈Aand x∈B}
={x∈Z|2≤x≤10 and 5 ≤x≤15}
={x∈Z|5≤x≤10}
={5,6,7,8,9,10}
Step 2: Find the union of sets Aand B.
A∪B={x∈Z|x∈Aor x∈B}
={x∈Z|2≤x≤10 or 5 ≤x≤15}
={x∈Z|2≤x≤15}
={2,3,4,5,6,7,8,9,10,11,12,13,14,15}
11
Question 21
Question
Let A,B, and Cbe sets such that |A∩B|= 12, |B∩C|= 15, |A∩C|= 20,
|A∩B∩C|= 7, |A∪B∪C|= 50, and |A∩B∩C′|= 5. Determine the number
of elements in |A∪B∪C′|.
Solution
Step 1: Use the principle of inclusion-exclusion to find the number of elements
in the union of the three sets A,B, and C.
|A∪B∪C|=|A|+|B|+|C|−|A∩B|−|B∩C|−|A∩C|+|A∩B∩C|
=|A|+|B|+|C| − 12 −15 −20 + 7
=|A|+|B|+|C| − 40
Step 2: Substituting the given information, we have
50 = |A|+|B|+|C| − 40
|A|+|B|+|C|= 90
Step 3: Use the fact |A∩B′∩C|=|A|−|A∩B|=|A| − 12 to determine
|A∩B′∩C|.
|A∩B′∩C|=|A|−|A∩B|
=|A| − 12
Step 4: Apply the principle of inclusion-exclusion to find the number of
elements in A∪B∪C′.
|A∪B∪C′|=|A|+|B|+|C|−|A∩B|−|B∩C|−|A∩C|+|A∩B∩C|+|A∩B′∩C|
= 90 −12 −15 −20 + 7 + (|A| − 12)
= 90 −40 −15 −20 −12 + 7
= 10
Therefore, the number of elements in |A∪B∪C′|is 10 .
Question 22
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|xis prime}. Determine the set
(A∩B)∪(A∩Bc).
12
Solution
Step 1: Find the intersection of set Aand set B.
A={x∈Z|1≤x≤10}
={1,2,3,4,5,6,7,8,9,10}
B={x∈Z|xis prime}
={2,3,5,7}
A∩B={2,3,5,7}
Step 2: Find the complement of set B. The complement of Bdenoted by Bc
is all integers between 1 and 10 that are not prime. This set is {1,4,6,8,9,10}.
Step 3: Find the intersection of set Aand set Bc.
A∩Bc={1,4,6,8,9,10}
Step 4: Combine the results from Step 1 and Step 3.
(A∩B)∪(A∩Bc) = {2,3,5,7}∪{1,4,6,8,9,10}
={1,2,3,4,5,6,7,8,9,10}
Therefore, (A∩B)∪(A∩Bc) = {1,2,3,4,5,6,7,8,9,10}.
Question 23
Question
Let A={x∈N|3≤x≤9}and B={x∈N|4≤x≤10}. Find A∩Band
A∪B.
Solution
Let’s first determine the elements in the sets Aand Bbefore finding their
intersection and union.
Step 1: Determine the elements in set ASet Acontains natural num-
bers xsuch that 3 ≤x≤9. Therefore, A={3,4,5,6,7,8,9}.
Step 2: Determine the elements in set BSet Bcontains natural num-
bers xsuch that 4 ≤x≤10. Therefore, B={4,5,6,7,8,9,10}.
Step 3: Find the intersection of sets Aand BThe intersection of
sets Aand B, denoted A∩B, contains elements that are present in both sets.
Therefore, A∩B={4,5,6,7,8,9}.
Step 4: Find the union of sets Aand BThe union of sets Aand B,
denoted A∪B, contains all elements that are present in either set. Therefore,
A∪B={3,4,5,6,7,8,9,10}.
13
Question 24
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|5≤x≤15}. Find
(A∪B)∩(AC∪BC).
Solution
Step 1: We first find A∪B.
A∪B={x∈Z|1≤x≤10}∪{x∈Z|5≤x≤15}
={x∈Z|1≤x≤15}
={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
Step 2: Next, we find ACand BC.
AC={x∈Z|x /∈A}={x∈Z|x≤0 or x > 10}
={x∈Z|x≤0}
={. . . , −3,−2,−1,0}
BC={x∈Z|x /∈B}={x∈Z|x < 5 or x > 15}
={x∈Z|x < 5}∪{x∈Z|x > 15}
={. . . , −3,−2,−1,0,1,2,3,4}∪{16,17,18,19, . . . }
Step 3: Now, we find (A∪B)∩(AC∪BC).
(A∪B)∩(AC∪BC) = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}∩
{. . . , −3,−2,−1,0,1,2,3,4}∪{16,17,18,19, . . . }
={1,2,3,4,5}
Therefore, (A∪B)∩(AC∪BC) = {1,2,3,4,5}.
Question 25
Question
Let A={x∈R|x2−3x+ 2 <0}and B={x∈R|2x−5<3}. Find A∩B.
Solution
Step 1: First, let’s find the values of xthat satisfy the inequality x2−3x+2 <0.
x2−3x+ 2 <0
(x−1)(x−2) <0
14
The solutions to the inequality x2−3x+ 2 <0 are 1 <x<2.
Step 2: Next, let’s find the values of xthat satisfy the inequality 2x−5<3.
2x−5<3
2x < 8
x < 4
The solutions to the inequality 2x−5<3 are x < 4.
Step 3: To find A∩B, we need to find the values of xthat satisfy both
1<x<2 and x < 4. The solutions that satisfy both inequalities are 1 <x<2.
Therefore, A∩B={x∈R|1<x<2}.
Question 26
Question
Let A={x∈Z|2< x ≤10}and B={x∈N|5< x ≤15}. Find the set
A∩B.
Solution
Step 1: Since A={x∈Z|2< x ≤10}, we have A={3,4,5,6,7,8,9,10}.
Step 2: Since B={x∈N|5< x ≤15}, we have B={6,7,8,9,10,11,12,13,14,15}.
Step 3: To find A∩B, we need to find the elements common to both sets A
and B.
Step 4: Therefore, A∩B={6,7,8,9,10}.
Hence, A∩B={6,7,8,9,10}.
Question 27
Question
Let A,B, and Cbe sets such that A∪B=A∪Cand A∩B=A∩C. Prove
that B=C.
Solution
We will prove that B=Cby showing that B⊆Cand C⊆B.
Step 1: Show that B⊆C
Assume x∈B. Since A∩B=A∩C, we have x∈Aif and only if x∈C.
Thus, x∈Bimplies x∈C, which shows that B⊆C.
Step 2: Show that C⊆B
Assume y∈C. Since A∪B=A∪C, we have y∈Aor y∈Bif and only
if y∈Aor y∈C. Since y∈A(from the fact that A∩B=A∩C), we can
conclude that y∈B, which shows that C⊆B.
Since we have shown that B⊆Cand C⊆B, we can conclude that B=C.
15
Question 28
Question
Let A,B, and Cbe sets such that A∪B=A∪Cand A∩B=A∩C. Prove
that B=C.
Solution
Step 1: Let’s start by proving that B⊆C. Since A∩B=A∩C, we have
A∩B⊆A∩Cby definition of subset. Then, for any x∈B, we have x∈A∩B,
which implies x∈A∩C. Thus, B⊆C.
Step 2: Next, we will prove that C⊆B. Since A∩B=A∩C, we have
A∩C⊆A∩Bby definition of subset. Then, for any y∈C, we have y∈A∩C,
which implies y∈A∩B. Thus, C⊆B.
Step 3: Combining Step 1 and Step 2, we conclude that B=Csince B⊆C
and C⊆B.
Question 29
Question
Let A={x∈Z: 2 ≤x≤8}and B={x∈Z: 5 ≤x≤12}. Find the set
A∪B.
Solution
Step 1: List out the elements of sets Aand Bindividually.
A={2,3,4,5,6,7,8}
B={5,6,7,8,9,10,11,12}
Step 2: Combine the elements of Aand Bto form the union A∪B.
,6,7,8, 9, 10, 11, 12}
={2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Therefore, A∪B={2,3,4,5,6,7,8,9,10,11,12}.
Question 30
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|6≤x≤15}. Find the set
A∪B.
16
Solution
Step 1: To find the intersection A∩B, we need to find the common elements in
both sets Aand B. Since Aconsists of solutions to the equation x2−3x−10 = 0,
we can factor it as (x−5)(x+ 2) = 0. Thus, A={−2,5}.
Similarly, Bconsists of solutions to the equation x2−4x+ 3 = 0, which
factors as (x−3)(x−1) = 0. Therefore, B={1,3}.
Step 2: Now, let’s find the intersection A∩Bby looking at the common
elements in Aand B. We see that A∩B={1,3}.
Step 3: Next, we will find the union A∪B, which consists of all unique
elements in sets Aand B. Therefore, A∪B={−2,1,3,5}.
Question 3
Question
Let A={x∈Z|2≤x≤10}and B={x∈Z|4≤x≤12}. Find the
intersection and union of sets Aand B.
Solution
Step 1: To find the intersection of sets Aand B, we need to determine the
elements that are common to both sets. Since A={2,3,4,5,6,7,8,9,10}
and B={4,5,6,7,8,9,10,11,12}, the intersection of Aand Bis A∩B=
{4,5,6,7,8,9,10}.
Step 2: To find the union of sets Aand B, we need to combine all elements
from both sets without repetitions. Since A={2,3,4,5,6,7,8,9,10}and B=
{4,5,6,7,8,9,10,11,12}, the union of Aand Bis A∪B={2,3,4,5,6,7,8,9,10,11,12}.
Question 4
Question
Let A={2,4,6,8,10}and B={3,6,9,12}. Define the set C=A∩Band
D=A∪B. Find the cardinality of set Cand set D.
Solution
Step 1: Find C=A∩B, the intersection of sets Aand B.
A∩B={x|x∈Aand x∈B}
={6}
Step 2: Find the cardinality of set C.
|C|=|{6}|
= 1
2
Step 3: Find D=A∪B, the union of sets Aand B.
A∪B={x|x∈Aor x∈B}
={2,3,4,6,8,9,10,12}
Step 4: Find the cardinality of set D.
|D|=|{2,3,4,6,8,9,10,12}|
= 8
Question 5
Question
Let A={x∈N|1≤x≤10}and B={x∈Z| −5≤x≤5}. Find the set
(A∩B)∪(A∪B).
Solution
Step 1: Find A∩B.
A∩B={x∈N|1≤x≤10}∩{x∈Z| −5≤x≤5}
={x∈Z|1≤x≤5}
Step 2: Find A∪B.
A∪B={x∈N|1≤x≤10}∪{x∈Z| −5≤x≤5}
={x∈Z| −5≤x≤10}
Step 3: Find (A∩B)∪(A∪B).
(A∩B)∪(A∪B) = {x∈Z|1≤x≤5}∪{x∈Z| −5≤x≤10}
={x∈Z| −5≤x≤10}
={x∈Z|x∈[−5,10]}
Therefore, (A∩B)∪(A∪B) = {x∈Z| −5≤x≤10}.
Question 6
Question
Let A={x|xis a prime number less than 10}and B={x|xis a positive even number less than 10}.
Determine the sets:
A∪Band A∩B
3
Solution
To find A∪B, we need to find all elements that are in either set Aor set B.
Step 1: Determine the elements of set A:
A={2,3,5,7}
Step 2: Determine the elements of set B:
B={2,4,6,8}
Step 3: Find the union of sets Aand B:
A∪B={2,3,4,5,6,7,8}
Now, to find A∩B, we need to find all elements that are common to both
set Aand set B.
Step 4: Find the intersection of sets Aand B:
A∩B={2}
Therefore, A∪B={2,3,4,5,6,7,8}and A∩B={2}.
Question 7
Question
Let A={x∈Z|3≤x≤10}and B={x∈Z|xis a prime number less than 15}.
Determine A∩B.
Solution
Step 1: Find the elements in set A. Since A={x∈Z|3≤x≤10}, the set A
contains integers from 3 to 10 inclusive. Thus, A={3,4,5,6,7,8,9,10}.
Step 2: Find the elements in set B. The prime numbers less than 15 are 2,
3, 5, 7, 11, and 13. Hence, B={2,3,5,7,11,13}.
Step 3: Determine A∩B. The intersection of sets Aand Bconsists of the
elements that are common in both Aand B. Therefore, A∩B={3,5,7}.
Question 8
Question
Let A,B, and Cbe sets such that A={x∈Z|1≤x≤10},B={x∈Z|5≤
x≤15}, and C={x∈Z|8≤x≤12}. Find (A∪B)∩C.
4
Solution
Step 1: We first find A∪B, the union of sets Aand B.
Let’s list the elements of set A:A={1,2,3,4,5,6,7,8,9,10}
Let’s list the elements of set B:B={5,6,7,8,9,10,11,12,13,14,15}
A∪B={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
Step 2: Next, we find (A∪B)∩C, the intersection of sets A∪Band C.
Let’s list the elements of set C:C={8,9,10,11,12}
(A∪B)∩C={8,9,10,11,12}
Thus, (A∪B)∩C={8,9,10,11,12}.
Question 9
Question
Let A={x∈N:xis a prime number less than 10}and B={1,2,3,4,5}.
Find A∩B.
Solution
To find the intersection A∩B, we need to determine the set of elements that
are common to both Aand B.
Step 1: Determine the prime numbers less than 10. The prime numbers
less than 10 are: 2, 3, 5, 7. Therefore, A={2,3,5,7}.
Step 2: Determine the intersection of Aand B. The intersection of Aand
Bis the set of elements that are common to both sets. In this case, A∩B=
{2,3,5}.
Therefore, A∩B={2,3,5}.
Question 10
Question
Let A={x∈Z|2≤x≤8}and B={x∈Z|4≤x≤10}. Find A∩Band
A∪B.
5
Solution
To find A∩B(the intersection of Aand B), weneedtodeterminetheelementsthatarecommontobothsetsAandB.
Step 1: List out the elements in sets Aand B.
A={2,3,4,5,6,7,8}
B={4,5,6,7,8,9,10}
Step 2: Identify the elements that are common to both sets.
A∩B={4,5,6,7,8}
Now, let’s find A∪B(the union of Aand B), whichcontainsalldistinctelementsfrombothsetsAandB.
Step 3: List out the elements in sets Aand B.
A={2,3,4,5,6,7,8}
B={4,5,6,7,8,9,10}
Step 4: Merge the elements from sets Aand B, removingduplicates.A∪B=
{2,3,4,5,6,7,8,9,10}
Therefore, A∩B={4,5,6,7,8}and A∪B={2,3,4,5,6,7,8,9,10}.
Question 11
Question
Let A={x∈Z|2≤x≤10}and B={x∈Z|5≤x≤15}. Find
(A∩B)∪(A\B).
Solution
Step 1: We first find A∩B, which represents the elements that are in both sets
Aand B.
A∩B={x∈Z|5≤x≤10}
Step 2: Next, we find A\B, which represents the elements that are in set
Abut not in set B.
A\B={x∈Z|2≤x < 5}
Step 3: Now, we find (A∩B)∪(A\B), which is the union of the elements
in A∩Band A\B.
(A∩B)∪(A\B) = {x∈Z|2≤x≤10 and x= 11,12,13,14,15}
Step 4: Simplifying the set obtained in the previous step, we have
(A∩B)∪(A\B) = {2,3,4,5,6,7,8,9,10}
Therefore, (A∩B)∪(A\B) = {2,3,4,5,6,7,8,9,10}.
6
Question 12
Question
Let Aand Bbe sets such that |A∪B|= 30, |A∩B|= 10, and |A−B|= 20.
Find |B|.
Solution
Step 1: We know that the number of elements in the union of two sets Aand
Bis given by the formula:
|A∪B|=|A|+|B|−|A∩B|.
Substitute the given values into this formula:
30 = |A|+|B| − 10.
Step 2: We also know that the number of elements in the set difference A−B
is given by the formula:
|A−B|=|A|−|A∩B|.
Substitute the given values into this formula:
20 = |A| − 10.
Step 3: Solve the system of equations from Step 1 and Step 2 to find |A|
and |B|. From Step 2, we have |A|= 30, and substituting this into the equation
from Step 1 gives:
30 = 30 + |B| − 10.
Step 4: Simplify the equation to solve for |B|:
|B|= 10.
Step 5: Therefore, the number of elements in set B, denoted by |B|, is 10.
Question 13
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|5≤x≤15}. Find the set
A∩B.
7
Solution
Step 1: To find the intersection A∩B, we need to determine the elements that
are common to both sets Aand B.
Step 2: Set Acontains the integers from 1 to 10, while set Bcontains the
integers from 5 to 15.
Step 3: The elements that are common to both sets are the integers that
satisfy both conditions. These integers are in the range from 5 to 10 since they
are present in both sets.
Step 4: Therefore, A∩B={x∈Z|5≤x≤10}.
Question 14
Question
Let A={x∈R|0< x ≤5}and B={x∈R|3<x<8}. Find the set A∩B.
Solution
Step 1: To find A∩B, we need to identify the elements that are common to
both sets Aand B.
Step 2: Set Aconsists of all real numbers greater than 0 and less than or
equal to 5, while set Bconsists of all real numbers greater than 3 and less than
8.
Step 3: We need to determine the intersection of sets Aand B, which includes
real numbers that satisfy both conditions (greater than 0 and less than or equal
to 5, and greater than 3 and less than 8).
Step 4: The intersection of sets Aand Bcan be defined as A∩B={x∈
R|3< x ≤5}.
Therefore, A∩B={x∈R|3< x ≤5}.
Question 15
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|5≤x≤15}. Determine the
following sets:
1. A∪B
2. A∩B
3. A\B
4. B\A
8
Solution
Given: A={x∈Z|1≤x≤10}B={x∈Z|5≤x≤15}
1. To find A∪B, we need to list all unique elements in Aand Bcombined:
A∪B={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
2. To find A∩B, we need to find the elements that are common in both A
and B:
A∩B={5,6,7,8,9,10}
3. To find A\B, we need to list the elements in Athat are not in B:
A\B={1,2,3,4}
4. To find B\A, we need to list the elements in Bthat are not in A:
B\A={11,12,13,14,15}
Question 16
Question
Let A={x∈Z|3≤x≤10}and B={x∈Z|5≤x≤12}. Find the set
A∩Band write the elements in ascending order.
Solution
Step 1: List the elements of Aand B.
A={3,4,5,6,7,8,9,10}
B={5,6,7,8,9,10,11,12}
Step 2: Find the intersection of sets Aand B.
A∩B={x|x∈Aand x∈B}
Step 3: Determine the common elements in both sets Aand B.
A∩B={5,6,7,8,9,10}
Step 4: Arrange the elements in ascending order.
A∩B={5,6,7,8,9,10}
Therefore, A∩B={5,6,7,8,9,10}.
9
Question 17
Question
Let A={x∈Z|0≤x≤10}and B={x∈Z|5≤x≤15}. Determine the
set A∩B.
Solution
Step 1: To find A∩B, we need to identify the elements that are present in both
sets Aand B.
Step 2: Set Acontains the integers from 0 to 10, while set Bcontains the
integers from 5 to 15.
Step 3: The intersection of Aand Bwill include the integers that are common
to both sets, namely the integers from 5 to 10 (inclusive).
Step 4: Therefore, the set A∩Bis {5,6,7,8,9,10}.
Question 18
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|xis prime}. Find the set
(A∪B)∩(A∩B).
Solution
Step 1: Let’s first determine the elements in set A:
A={1,2,3,4,5,6,7,8,9,10}
Step 2: Next, we determine the elements in set B:
B={2,3,5,7}
Step 3: Now, we find the union of sets Aand B:
A∪B={1,2,3,4,5,6,7,8,9,10}∪{2,3,5,7}={1,2,3,4,5,6,7,8,9,10}
Step 4: Next, we find the intersection of sets Aand B:
A∩B={1,2,3,4,5,6,7,8,9,10}∩{2,3,5,7}={2,3,5,7}
Step 5: Finally, we find the intersection of (A∪B) and (A∩B):
(A∪B)∩(A∩B) = {1,2,3,4,5,6,7,8,9,10}∩{2,3,5,7}={2,3,5,7}
Therefore, the set (A∪B)∩(A∩B) is {2,3,5,7}.
10
Question 19
Question
Let A={x∈Z|1≤x≤10},B={x∈Z|5≤x≤15}, and C={x∈Z|
xis prime}. Find (A∪B)−C.
Solution
Step 1: Find A∪B
A∪B={x∈Z|1≤x≤10}∪{x∈Z|5≤x≤15}
={x∈Z|1≤x≤15}
Step 2: Find (A∪B)−C
(A∪B)−C={x∈Z|1≤x≤15}−{x∈Z|xis prime}
={2,4,6,8,10,12,14}
Therefore, (A∪B)−C={2,4,6,8,10,12,14}.
Question 20
Question
Let Aand Bbe sets such that A={x∈Z|2≤x≤10}and B={x∈Z|5≤
x≤15}. Find the intersection and union of sets Aand B.
Solution
Step 1: Find the intersection of sets Aand B.
A∩B={x∈Z|x∈Aand x∈B}
={x∈Z|2≤x≤10 and 5 ≤x≤15}
={x∈Z|5≤x≤10}
={5,6,7,8,9,10}
Step 2: Find the union of sets Aand B.
A∪B={x∈Z|x∈Aor x∈B}
={x∈Z|2≤x≤10 or 5 ≤x≤15}
={x∈Z|2≤x≤15}
={2,3,4,5,6,7,8,9,10,11,12,13,14,15}
11
Question 21
Question
Let A,B, and Cbe sets such that |A∩B|= 12, |B∩C|= 15, |A∩C|= 20,
|A∩B∩C|= 7, |A∪B∪C|= 50, and |A∩B∩C′|= 5. Determine the number
of elements in |A∪B∪C′|.
Solution
Step 1: Use the principle of inclusion-exclusion to find the number of elements
in the union of the three sets A,B, and C.
|A∪B∪C|=|A|+|B|+|C|−|A∩B|−|B∩C|−|A∩C|+|A∩B∩C|
=|A|+|B|+|C| − 12 −15 −20 + 7
=|A|+|B|+|C| − 40
Step 2: Substituting the given information, we have
50 = |A|+|B|+|C| − 40
|A|+|B|+|C|= 90
Step 3: Use the fact |A∩B′∩C|=|A|−|A∩B|=|A| − 12 to determine
|A∩B′∩C|.
|A∩B′∩C|=|A|−|A∩B|
=|A| − 12
Step 4: Apply the principle of inclusion-exclusion to find the number of
elements in A∪B∪C′.
|A∪B∪C′|=|A|+|B|+|C|−|A∩B|−|B∩C|−|A∩C|+|A∩B∩C|+|A∩B′∩C|
= 90 −12 −15 −20 + 7 + (|A| − 12)
= 90 −40 −15 −20 −12 + 7
= 10
Therefore, the number of elements in |A∪B∪C′|is 10 .
Question 22
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|xis prime}. Determine the set
(A∩B)∪(A∩Bc).
12
Solution
Step 1: Find the intersection of set Aand set B.
A={x∈Z|1≤x≤10}
={1,2,3,4,5,6,7,8,9,10}
B={x∈Z|xis prime}
={2,3,5,7}
A∩B={2,3,5,7}
Step 2: Find the complement of set B. The complement of Bdenoted by Bc
is all integers between 1 and 10 that are not prime. This set is {1,4,6,8,9,10}.
Step 3: Find the intersection of set Aand set Bc.
A∩Bc={1,4,6,8,9,10}
Step 4: Combine the results from Step 1 and Step 3.
(A∩B)∪(A∩Bc) = {2,3,5,7}∪{1,4,6,8,9,10}
={1,2,3,4,5,6,7,8,9,10}
Therefore, (A∩B)∪(A∩Bc) = {1,2,3,4,5,6,7,8,9,10}.
Question 23
Question
Let A={x∈N|3≤x≤9}and B={x∈N|4≤x≤10}. Find A∩Band
A∪B.
Solution
Let’s first determine the elements in the sets Aand Bbefore finding their
intersection and union.
Step 1: Determine the elements in set ASet Acontains natural num-
bers xsuch that 3 ≤x≤9. Therefore, A={3,4,5,6,7,8,9}.
Step 2: Determine the elements in set BSet Bcontains natural num-
bers xsuch that 4 ≤x≤10. Therefore, B={4,5,6,7,8,9,10}.
Step 3: Find the intersection of sets Aand BThe intersection of
sets Aand B, denoted A∩B, contains elements that are present in both sets.
Therefore, A∩B={4,5,6,7,8,9}.
Step 4: Find the union of sets Aand BThe union of sets Aand B,
denoted A∪B, contains all elements that are present in either set. Therefore,
A∪B={3,4,5,6,7,8,9,10}.
13
Question 24
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|5≤x≤15}. Find
(A∪B)∩(AC∪BC).
Solution
Step 1: We first find A∪B.
A∪B={x∈Z|1≤x≤10}∪{x∈Z|5≤x≤15}
={x∈Z|1≤x≤15}
={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
Step 2: Next, we find ACand BC.
AC={x∈Z|x /∈A}={x∈Z|x≤0 or x > 10}
={x∈Z|x≤0}
={. . . , −3,−2,−1,0}
BC={x∈Z|x /∈B}={x∈Z|x < 5 or x > 15}
={x∈Z|x < 5}∪{x∈Z|x > 15}
={. . . , −3,−2,−1,0,1,2,3,4}∪{16,17,18,19, . . . }
Step 3: Now, we find (A∪B)∩(AC∪BC).
(A∪B)∩(AC∪BC) = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}∩
{. . . , −3,−2,−1,0,1,2,3,4}∪{16,17,18,19, . . . }
={1,2,3,4,5}
Therefore, (A∪B)∩(AC∪BC) = {1,2,3,4,5}.
Question 25
Question
Let A={x∈R|x2−3x+ 2 <0}and B={x∈R|2x−5<3}. Find A∩B.
Solution
Step 1: First, let’s find the values of xthat satisfy the inequality x2−3x+2 <0.
x2−3x+ 2 <0
(x−1)(x−2) <0
14
The solutions to the inequality x2−3x+ 2 <0 are 1 <x<2.
Step 2: Next, let’s find the values of xthat satisfy the inequality 2x−5<3.
2x−5<3
2x < 8
x < 4
The solutions to the inequality 2x−5<3 are x < 4.
Step 3: To find A∩B, we need to find the values of xthat satisfy both
1<x<2 and x < 4. The solutions that satisfy both inequalities are 1 <x<2.
Therefore, A∩B={x∈R|1<x<2}.
Question 26
Question
Let A={x∈Z|2< x ≤10}and B={x∈N|5< x ≤15}. Find the set
A∩B.
Solution
Step 1: Since A={x∈Z|2< x ≤10}, we have A={3,4,5,6,7,8,9,10}.
Step 2: Since B={x∈N|5< x ≤15}, we have B={6,7,8,9,10,11,12,13,14,15}.
Step 3: To find A∩B, we need to find the elements common to both sets A
and B.
Step 4: Therefore, A∩B={6,7,8,9,10}.
Hence, A∩B={6,7,8,9,10}.
Question 27
Question
Let A,B, and Cbe sets such that A∪B=A∪Cand A∩B=A∩C. Prove
that B=C.
Solution
We will prove that B=Cby showing that B⊆Cand C⊆B.
Step 1: Show that B⊆C
Assume x∈B. Since A∩B=A∩C, we have x∈Aif and only if x∈C.
Thus, x∈Bimplies x∈C, which shows that B⊆C.
Step 2: Show that C⊆B
Assume y∈C. Since A∪B=A∪C, we have y∈Aor y∈Bif and only
if y∈Aor y∈C. Since y∈A(from the fact that A∩B=A∩C), we can
conclude that y∈B, which shows that C⊆B.
Since we have shown that B⊆Cand C⊆B, we can conclude that B=C.
15
Question 28
Question
Let A,B, and Cbe sets such that A∪B=A∪Cand A∩B=A∩C. Prove
that B=C.
Solution
Step 1: Let’s start by proving that B⊆C. Since A∩B=A∩C, we have
A∩B⊆A∩Cby definition of subset. Then, for any x∈B, we have x∈A∩B,
which implies x∈A∩C. Thus, B⊆C.
Step 2: Next, we will prove that C⊆B. Since A∩B=A∩C, we have
A∩C⊆A∩Bby definition of subset. Then, for any y∈C, we have y∈A∩C,
which implies y∈A∩B. Thus, C⊆B.
Step 3: Combining Step 1 and Step 2, we conclude that B=Csince B⊆C
and C⊆B.
Question 29
Question
Let A={x∈Z: 2 ≤x≤8}and B={x∈Z: 5 ≤x≤12}. Find the set
A∪B.
Solution
Step 1: List out the elements of sets Aand Bindividually.
A={2,3,4,5,6,7,8}
B={5,6,7,8,9,10,11,12}
Step 2: Combine the elements of Aand Bto form the union A∪B.
,6,7,8, 9, 10, 11, 12}
={2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Therefore, A∪B={2,3,4,5,6,7,8,9,10,11,12}.
Question 30
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|6≤x≤15}. Find the set
A∪B.
16
Solution
Step 1: To find the intersection A∩B, we need to find the common elements in
both sets Aand B. Since Aconsists of solutions to the equation x2−3x−10 = 0,
we can factor it as (x−5)(x+ 2) = 0. Thus, A={−2,5}.
Similarly, Bconsists of solutions to the equation x2−4x+ 3 = 0, which
factors as (x−3)(x−1) = 0. Therefore, B={1,3}.
Step 2: Now, let’s find the intersection A∩Bby looking at the common
elements in Aand B. We see that A∩B={1,3}.
Step 3: Next, we will find the union A∪B, which consists of all unique
elements in sets Aand B. Therefore, A∪B={−2,1,3,5}.
Question 3
Question
Let A={x∈Z|2≤x≤10}and B={x∈Z|4≤x≤12}. Find the
intersection and union of sets Aand B.
Solution
Step 1: To find the intersection of sets Aand B, we need to determine the
elements that are common to both sets. Since A={2,3,4,5,6,7,8,9,10}
and B={4,5,6,7,8,9,10,11,12}, the intersection of Aand Bis A∩B=
{4,5,6,7,8,9,10}.
Step 2: To find the union of sets Aand B, we need to combine all elements
from both sets without repetitions. Since A={2,3,4,5,6,7,8,9,10}and B=
{4,5,6,7,8,9,10,11,12}, the union of Aand Bis A∪B={2,3,4,5,6,7,8,9,10,11,12}.
Question 4
Question
Let A={2,4,6,8,10}and B={3,6,9,12}. Define the set C=A∩Band
D=A∪B. Find the cardinality of set Cand set D.
Solution
Step 1: Find C=A∩B, the intersection of sets Aand B.
A∩B={x|x∈Aand x∈B}
={6}
Step 2: Find the cardinality of set C.
|C|=|{6}|
= 1
2
Step 3: Find D=A∪B, the union of sets Aand B.
A∪B={x|x∈Aor x∈B}
={2,3,4,6,8,9,10,12}
Step 4: Find the cardinality of set D.
|D|=|{2,3,4,6,8,9,10,12}|
= 8
Question 5
Question
Let A={x∈N|1≤x≤10}and B={x∈Z| −5≤x≤5}. Find the set
(A∩B)∪(A∪B).
Solution
Step 1: Find A∩B.
A∩B={x∈N|1≤x≤10}∩{x∈Z| −5≤x≤5}
={x∈Z|1≤x≤5}
Step 2: Find A∪B.
A∪B={x∈N|1≤x≤10}∪{x∈Z| −5≤x≤5}
={x∈Z| −5≤x≤10}
Step 3: Find (A∩B)∪(A∪B).
(A∩B)∪(A∪B) = {x∈Z|1≤x≤5}∪{x∈Z| −5≤x≤10}
={x∈Z| −5≤x≤10}
={x∈Z|x∈[−5,10]}
Therefore, (A∩B)∪(A∪B) = {x∈Z| −5≤x≤10}.
Question 6
Question
Let A={x|xis a prime number less than 10}and B={x|xis a positive even number less than 10}.
Determine the sets:
A∪Band A∩B
3
Solution
To find A∪B, we need to find all elements that are in either set Aor set B.
Step 1: Determine the elements of set A:
A={2,3,5,7}
Step 2: Determine the elements of set B:
B={2,4,6,8}
Step 3: Find the union of sets Aand B:
A∪B={2,3,4,5,6,7,8}
Now, to find A∩B, we need to find all elements that are common to both
set Aand set B.
Step 4: Find the intersection of sets Aand B:
A∩B={2}
Therefore, A∪B={2,3,4,5,6,7,8}and A∩B={2}.
Question 7
Question
Let A={x∈Z|3≤x≤10}and B={x∈Z|xis a prime number less than 15}.
Determine A∩B.
Solution
Step 1: Find the elements in set A. Since A={x∈Z|3≤x≤10}, the set A
contains integers from 3 to 10 inclusive. Thus, A={3,4,5,6,7,8,9,10}.
Step 2: Find the elements in set B. The prime numbers less than 15 are 2,
3, 5, 7, 11, and 13. Hence, B={2,3,5,7,11,13}.
Step 3: Determine A∩B. The intersection of sets Aand Bconsists of the
elements that are common in both Aand B. Therefore, A∩B={3,5,7}.
Question 8
Question
Let A,B, and Cbe sets such that A={x∈Z|1≤x≤10},B={x∈Z|5≤
x≤15}, and C={x∈Z|8≤x≤12}. Find (A∪B)∩C.
4
Solution
Step 1: We first find A∪B, the union of sets Aand B.
Let’s list the elements of set A:A={1,2,3,4,5,6,7,8,9,10}
Let’s list the elements of set B:B={5,6,7,8,9,10,11,12,13,14,15}
A∪B={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
Step 2: Next, we find (A∪B)∩C, the intersection of sets A∪Band C.
Let’s list the elements of set C:C={8,9,10,11,12}
(A∪B)∩C={8,9,10,11,12}
Thus, (A∪B)∩C={8,9,10,11,12}.
Question 9
Question
Let A={x∈N:xis a prime number less than 10}and B={1,2,3,4,5}.
Find A∩B.
Solution
To find the intersection A∩B, we need to determine the set of elements that
are common to both Aand B.
Step 1: Determine the prime numbers less than 10. The prime numbers
less than 10 are: 2, 3, 5, 7. Therefore, A={2,3,5,7}.
Step 2: Determine the intersection of Aand B. The intersection of Aand
Bis the set of elements that are common to both sets. In this case, A∩B=
{2,3,5}.
Therefore, A∩B={2,3,5}.
Question 10
Question
Let A={x∈Z|2≤x≤8}and B={x∈Z|4≤x≤10}. Find A∩Band
A∪B.
5
Solution
To find A∩B(the intersection of Aand B), weneedtodeterminetheelementsthatarecommontobothsetsAandB.
Step 1: List out the elements in sets Aand B.
A={2,3,4,5,6,7,8}
B={4,5,6,7,8,9,10}
Step 2: Identify the elements that are common to both sets.
A∩B={4,5,6,7,8}
Now, let’s find A∪B(the union of Aand B), whichcontainsalldistinctelementsfrombothsetsAandB.
Step 3: List out the elements in sets Aand B.
A={2,3,4,5,6,7,8}
B={4,5,6,7,8,9,10}
Step 4: Merge the elements from sets Aand B, removingduplicates.A∪B=
{2,3,4,5,6,7,8,9,10}
Therefore, A∩B={4,5,6,7,8}and A∪B={2,3,4,5,6,7,8,9,10}.
Question 11
Question
Let A={x∈Z|2≤x≤10}and B={x∈Z|5≤x≤15}. Find
(A∩B)∪(A\B).
Solution
Step 1: We first find A∩B, which represents the elements that are in both sets
Aand B.
A∩B={x∈Z|5≤x≤10}
Step 2: Next, we find A\B, which represents the elements that are in set
Abut not in set B.
A\B={x∈Z|2≤x < 5}
Step 3: Now, we find (A∩B)∪(A\B), which is the union of the elements
in A∩Band A\B.
(A∩B)∪(A\B) = {x∈Z|2≤x≤10 and x= 11,12,13,14,15}
Step 4: Simplifying the set obtained in the previous step, we have
(A∩B)∪(A\B) = {2,3,4,5,6,7,8,9,10}
Therefore, (A∩B)∪(A\B) = {2,3,4,5,6,7,8,9,10}.
6
Question 12
Question
Let Aand Bbe sets such that |A∪B|= 30, |A∩B|= 10, and |A−B|= 20.
Find |B|.
Solution
Step 1: We know that the number of elements in the union of two sets Aand
Bis given by the formula:
|A∪B|=|A|+|B|−|A∩B|.
Substitute the given values into this formula:
30 = |A|+|B| − 10.
Step 2: We also know that the number of elements in the set difference A−B
is given by the formula:
|A−B|=|A|−|A∩B|.
Substitute the given values into this formula:
20 = |A| − 10.
Step 3: Solve the system of equations from Step 1 and Step 2 to find |A|
and |B|. From Step 2, we have |A|= 30, and substituting this into the equation
from Step 1 gives:
30 = 30 + |B| − 10.
Step 4: Simplify the equation to solve for |B|:
|B|= 10.
Step 5: Therefore, the number of elements in set B, denoted by |B|, is 10.
Question 13
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|5≤x≤15}. Find the set
A∩B.
7
Solution
Step 1: To find the intersection A∩B, we need to determine the elements that
are common to both sets Aand B.
Step 2: Set Acontains the integers from 1 to 10, while set Bcontains the
integers from 5 to 15.
Step 3: The elements that are common to both sets are the integers that
satisfy both conditions. These integers are in the range from 5 to 10 since they
are present in both sets.
Step 4: Therefore, A∩B={x∈Z|5≤x≤10}.
Question 14
Question
Let A={x∈R|0< x ≤5}and B={x∈R|3<x<8}. Find the set A∩B.
Solution
Step 1: To find A∩B, we need to identify the elements that are common to
both sets Aand B.
Step 2: Set Aconsists of all real numbers greater than 0 and less than or
equal to 5, while set Bconsists of all real numbers greater than 3 and less than
8.
Step 3: We need to determine the intersection of sets Aand B, which includes
real numbers that satisfy both conditions (greater than 0 and less than or equal
to 5, and greater than 3 and less than 8).
Step 4: The intersection of sets Aand Bcan be defined as A∩B={x∈
R|3< x ≤5}.
Therefore, A∩B={x∈R|3< x ≤5}.
Question 15
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|5≤x≤15}. Determine the
following sets:
1. A∪B
2. A∩B
3. A\B
4. B\A
8
Solution
Given: A={x∈Z|1≤x≤10}B={x∈Z|5≤x≤15}
1. To find A∪B, we need to list all unique elements in Aand Bcombined:
A∪B={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
2. To find A∩B, we need to find the elements that are common in both A
and B:
A∩B={5,6,7,8,9,10}
3. To find A\B, we need to list the elements in Athat are not in B:
A\B={1,2,3,4}
4. To find B\A, we need to list the elements in Bthat are not in A:
B\A={11,12,13,14,15}
Question 16
Question
Let A={x∈Z|3≤x≤10}and B={x∈Z|5≤x≤12}. Find the set
A∩Band write the elements in ascending order.
Solution
Step 1: List the elements of Aand B.
A={3,4,5,6,7,8,9,10}
B={5,6,7,8,9,10,11,12}
Step 2: Find the intersection of sets Aand B.
A∩B={x|x∈Aand x∈B}
Step 3: Determine the common elements in both sets Aand B.
A∩B={5,6,7,8,9,10}
Step 4: Arrange the elements in ascending order.
A∩B={5,6,7,8,9,10}
Therefore, A∩B={5,6,7,8,9,10}.
9
Question 17
Question
Let A={x∈Z|0≤x≤10}and B={x∈Z|5≤x≤15}. Determine the
set A∩B.
Solution
Step 1: To find A∩B, we need to identify the elements that are present in both
sets Aand B.
Step 2: Set Acontains the integers from 0 to 10, while set Bcontains the
integers from 5 to 15.
Step 3: The intersection of Aand Bwill include the integers that are common
to both sets, namely the integers from 5 to 10 (inclusive).
Step 4: Therefore, the set A∩Bis {5,6,7,8,9,10}.
Question 18
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|xis prime}. Find the set
(A∪B)∩(A∩B).
Solution
Step 1: Let’s first determine the elements in set A:
A={1,2,3,4,5,6,7,8,9,10}
Step 2: Next, we determine the elements in set B:
B={2,3,5,7}
Step 3: Now, we find the union of sets Aand B:
A∪B={1,2,3,4,5,6,7,8,9,10}∪{2,3,5,7}={1,2,3,4,5,6,7,8,9,10}
Step 4: Next, we find the intersection of sets Aand B:
A∩B={1,2,3,4,5,6,7,8,9,10}∩{2,3,5,7}={2,3,5,7}
Step 5: Finally, we find the intersection of (A∪B) and (A∩B):
(A∪B)∩(A∩B) = {1,2,3,4,5,6,7,8,9,10}∩{2,3,5,7}={2,3,5,7}
Therefore, the set (A∪B)∩(A∩B) is {2,3,5,7}.
10
Question 19
Question
Let A={x∈Z|1≤x≤10},B={x∈Z|5≤x≤15}, and C={x∈Z|
xis prime}. Find (A∪B)−C.
Solution
Step 1: Find A∪B
A∪B={x∈Z|1≤x≤10}∪{x∈Z|5≤x≤15}
={x∈Z|1≤x≤15}
Step 2: Find (A∪B)−C
(A∪B)−C={x∈Z|1≤x≤15}−{x∈Z|xis prime}
={2,4,6,8,10,12,14}
Therefore, (A∪B)−C={2,4,6,8,10,12,14}.
Question 20
Question
Let Aand Bbe sets such that A={x∈Z|2≤x≤10}and B={x∈Z|5≤
x≤15}. Find the intersection and union of sets Aand B.
Solution
Step 1: Find the intersection of sets Aand B.
A∩B={x∈Z|x∈Aand x∈B}
={x∈Z|2≤x≤10 and 5 ≤x≤15}
={x∈Z|5≤x≤10}
={5,6,7,8,9,10}
Step 2: Find the union of sets Aand B.
A∪B={x∈Z|x∈Aor x∈B}
={x∈Z|2≤x≤10 or 5 ≤x≤15}
={x∈Z|2≤x≤15}
={2,3,4,5,6,7,8,9,10,11,12,13,14,15}
11
Question 21
Question
Let A,B, and Cbe sets such that |A∩B|= 12, |B∩C|= 15, |A∩C|= 20,
|A∩B∩C|= 7, |A∪B∪C|= 50, and |A∩B∩C′|= 5. Determine the number
of elements in |A∪B∪C′|.
Solution
Step 1: Use the principle of inclusion-exclusion to find the number of elements
in the union of the three sets A,B, and C.
|A∪B∪C|=|A|+|B|+|C|−|A∩B|−|B∩C|−|A∩C|+|A∩B∩C|
=|A|+|B|+|C| − 12 −15 −20 + 7
=|A|+|B|+|C| − 40
Step 2: Substituting the given information, we have
50 = |A|+|B|+|C| − 40
|A|+|B|+|C|= 90
Step 3: Use the fact |A∩B′∩C|=|A|−|A∩B|=|A| − 12 to determine
|A∩B′∩C|.
|A∩B′∩C|=|A|−|A∩B|
=|A| − 12
Step 4: Apply the principle of inclusion-exclusion to find the number of
elements in A∪B∪C′.
|A∪B∪C′|=|A|+|B|+|C|−|A∩B|−|B∩C|−|A∩C|+|A∩B∩C|+|A∩B′∩C|
= 90 −12 −15 −20 + 7 + (|A| − 12)
= 90 −40 −15 −20 −12 + 7
= 10
Therefore, the number of elements in |A∪B∪C′|is 10 .
Question 22
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|xis prime}. Determine the set
(A∩B)∪(A∩Bc).
12
Solution
Step 1: Find the intersection of set Aand set B.
A={x∈Z|1≤x≤10}
={1,2,3,4,5,6,7,8,9,10}
B={x∈Z|xis prime}
={2,3,5,7}
A∩B={2,3,5,7}
Step 2: Find the complement of set B. The complement of Bdenoted by Bc
is all integers between 1 and 10 that are not prime. This set is {1,4,6,8,9,10}.
Step 3: Find the intersection of set Aand set Bc.
A∩Bc={1,4,6,8,9,10}
Step 4: Combine the results from Step 1 and Step 3.
(A∩B)∪(A∩Bc) = {2,3,5,7}∪{1,4,6,8,9,10}
={1,2,3,4,5,6,7,8,9,10}
Therefore, (A∩B)∪(A∩Bc) = {1,2,3,4,5,6,7,8,9,10}.
Question 23
Question
Let A={x∈N|3≤x≤9}and B={x∈N|4≤x≤10}. Find A∩Band
A∪B.
Solution
Let’s first determine the elements in the sets Aand Bbefore finding their
intersection and union.
Step 1: Determine the elements in set ASet Acontains natural num-
bers xsuch that 3 ≤x≤9. Therefore, A={3,4,5,6,7,8,9}.
Step 2: Determine the elements in set BSet Bcontains natural num-
bers xsuch that 4 ≤x≤10. Therefore, B={4,5,6,7,8,9,10}.
Step 3: Find the intersection of sets Aand BThe intersection of
sets Aand B, denoted A∩B, contains elements that are present in both sets.
Therefore, A∩B={4,5,6,7,8,9}.
Step 4: Find the union of sets Aand BThe union of sets Aand B,
denoted A∪B, contains all elements that are present in either set. Therefore,
A∪B={3,4,5,6,7,8,9,10}.
13
Question 24
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|5≤x≤15}. Find
(A∪B)∩(AC∪BC).
Solution
Step 1: We first find A∪B.
A∪B={x∈Z|1≤x≤10}∪{x∈Z|5≤x≤15}
={x∈Z|1≤x≤15}
={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
Step 2: Next, we find ACand BC.
AC={x∈Z|x /∈A}={x∈Z|x≤0 or x > 10}
={x∈Z|x≤0}
={. . . , −3,−2,−1,0}
BC={x∈Z|x /∈B}={x∈Z|x < 5 or x > 15}
={x∈Z|x < 5}∪{x∈Z|x > 15}
={. . . , −3,−2,−1,0,1,2,3,4}∪{16,17,18,19, . . . }
Step 3: Now, we find (A∪B)∩(AC∪BC).
(A∪B)∩(AC∪BC) = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}∩
{. . . , −3,−2,−1,0,1,2,3,4}∪{16,17,18,19, . . . }
={1,2,3,4,5}
Therefore, (A∪B)∩(AC∪BC) = {1,2,3,4,5}.
Question 25
Question
Let A={x∈R|x2−3x+ 2 <0}and B={x∈R|2x−5<3}. Find A∩B.
Solution
Step 1: First, let’s find the values of xthat satisfy the inequality x2−3x+2 <0.
x2−3x+ 2 <0
(x−1)(x−2) <0
14
The solutions to the inequality x2−3x+ 2 <0 are 1 <x<2.
Step 2: Next, let’s find the values of xthat satisfy the inequality 2x−5<3.
2x−5<3
2x < 8
x < 4
The solutions to the inequality 2x−5<3 are x < 4.
Step 3: To find A∩B, we need to find the values of xthat satisfy both
1<x<2 and x < 4. The solutions that satisfy both inequalities are 1 <x<2.
Therefore, A∩B={x∈R|1<x<2}.
Question 26
Question
Let A={x∈Z|2< x ≤10}and B={x∈N|5< x ≤15}. Find the set
A∩B.
Solution
Step 1: Since A={x∈Z|2< x ≤10}, we have A={3,4,5,6,7,8,9,10}.
Step 2: Since B={x∈N|5< x ≤15}, we have B={6,7,8,9,10,11,12,13,14,15}.
Step 3: To find A∩B, we need to find the elements common to both sets A
and B.
Step 4: Therefore, A∩B={6,7,8,9,10}.
Hence, A∩B={6,7,8,9,10}.
Question 27
Question
Let A,B, and Cbe sets such that A∪B=A∪Cand A∩B=A∩C. Prove
that B=C.
Solution
We will prove that B=Cby showing that B⊆Cand C⊆B.
Step 1: Show that B⊆C
Assume x∈B. Since A∩B=A∩C, we have x∈Aif and only if x∈C.
Thus, x∈Bimplies x∈C, which shows that B⊆C.
Step 2: Show that C⊆B
Assume y∈C. Since A∪B=A∪C, we have y∈Aor y∈Bif and only
if y∈Aor y∈C. Since y∈A(from the fact that A∩B=A∩C), we can
conclude that y∈B, which shows that C⊆B.
Since we have shown that B⊆Cand C⊆B, we can conclude that B=C.
15
Question 28
Question
Let A,B, and Cbe sets such that A∪B=A∪Cand A∩B=A∩C. Prove
that B=C.
Solution
Step 1: Let’s start by proving that B⊆C. Since A∩B=A∩C, we have
A∩B⊆A∩Cby definition of subset. Then, for any x∈B, we have x∈A∩B,
which implies x∈A∩C. Thus, B⊆C.
Step 2: Next, we will prove that C⊆B. Since A∩B=A∩C, we have
A∩C⊆A∩Bby definition of subset. Then, for any y∈C, we have y∈A∩C,
which implies y∈A∩B. Thus, C⊆B.
Step 3: Combining Step 1 and Step 2, we conclude that B=Csince B⊆C
and C⊆B.
Question 29
Question
Let A={x∈Z: 2 ≤x≤8}and B={x∈Z: 5 ≤x≤12}. Find the set
A∪B.
Solution
Step 1: List out the elements of sets Aand Bindividually.
A={2,3,4,5,6,7,8}
B={5,6,7,8,9,10,11,12}
Step 2: Combine the elements of Aand Bto form the union A∪B.
,6,7,8, 9, 10, 11, 12}
={2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Therefore, A∪B={2,3,4,5,6,7,8,9,10,11,12}.
Question 30
Question
Let A={x∈Z|1≤x≤10}and B={x∈Z|6≤x≤15}. Find the set
A∪B.
16
Solution
Step 1: We first list out the elements of set Aand set B.
A={1,2,3,4,5,6,7,8,9,10}
B={6,7,8,9,10,11,12,13,14,15}
Step 2: To find A∪B, we look at the elements that are present in either set
Aor set B(or both).
A∪B={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
Therefore, A∪B={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}.
17
