Chapter 2, problem 2: (1 pts) Show truth tables for each of the following. + 1
b) The system has four inputs and three outputs. The first two inputs, and , represent a 2-bit a b
binary number (range of 0 to 3). A second binary number (same range) is represented by the
other two inputs, and . The output is to be 1 if and only if the two numbers differ by exactly c d f
2. Output is to be 1 if and only if the numbers are equal. Output is to be 1 if and only of the g h
second number is larger than the first.
Answer:
a b c d f g h
0 0 0 0 0 1 0
0 0 0 1 0 0 1
0 0 1 0 1 0 1
0 0 1 1 0 0 1
0 1 0 0 0 0 0
0 1 0 1 0 1 0
0 1 1 0 0 0 1
0 1 1 1 1 0 1
1 0 0 0 1 0 0
1 0 0 1 0 0 0
1 0 1 0 0 1 0
1 0 1 1 0 0 1
1 1 0 0 0 0 0
1 1 0 1 1 0 0
1 1 1 0 0 0 0
1 1 1 1 0 1 0
f) The system has four inputs. The first two, and , represent a number in the range 0 to 2 (3 is a b
not used). The other two, and , represent a second number in the same range. The output, , is c d y
to be 1 if and only if the two numbers do not differ by more than 1.
Answer:
a
b
c
d
y
0
0
0
0
1
0
0
0
1
1
0
0
1
0
0
0
0
1
1
X
0
1
0
0
1
0
1
0
1
1
0
1
1
0
1
0
1
1
1
X
1
0
0
0
0
1
0
0
1
1
1
0
1
0
1
1
0
1
1
X
1
1
0
0
X
1
1
0
1
X
1
1
1
0
X
1
1
1
1
X
Chapter 2, problem 4: (1 pts) Show a truth table for the following function:
a) F = X'Y + Y'Z' + XYZ
Answer:
X Y Z X’Y Y’Z’ XYZ Output
0 0 0 0 1 0 1
0 0 1 0 0 0 0
0 1 0 1 0 0 1
1 0 0 0 1 0 1
0 1 1 1 0 0 1
1 1 0 0 0 0 0
1 0 1 0 0 0 0
1 1 1 0 0 1 1
Chapter 2, problem 8: (1 pts) Using Boolean Algebra, reduce the following expression to a
minimum sum of products form.
d) a’b’c’ + a’b’c + abc + ab’c
Answer: a’b’ + ac
Chapter 2, problem 10: (1 pts) Show a block diagram of a system using AND, OR, and NOT
gates to implement the following function. Assume that variables are available only
uncomplemented. Do not manipulate the algebra.
a) P'Q' + PR + Q'R
Chapter 2, problem 11: (1 pts) Express the following circuit in sum of product form.
Answer:
f = a’bc + b’d + ac’
Chapter 2, problem 14: (6 pts) For the function in the following truth table: f
a b c f
0 0 0 0 0
1 0 0 1 1
a) Show the minterms in numerical form.
f = ∑m(1, 5, 6, 9)
b) Show the canonical algebraic expression in sum of products form.
f = a’b’c + ab’c + abc’ + abc
c) Show a minimum SOP expression.
f = b’c + ab
d) Show the minterms of f’ in numeric form.
f’ = ∑m(0, 2, 3, 4)
e) Show the canonical algebraic expression in product of sums form.
f = (a + b + c) (a + b’ + c) (a + b’ + c’) (a’ + b + c)
f) Show a minimum POS expression ( : 2 terms, 4 literals). f
f = (a + b’) (c + b)
2 0 1 0 0
3 0 1 1 0
4 1 0 0 0
5 1 0 1 1
6 1 1 0 1
7 1 1 1 1
Chapter 2, problem 15: (6 pts) For each of the following functions:
F = AB' + BC + AC
G = (A + B)(A + C') + AB'
a) Show the truth table.
Answer:
A
B
C
AB’
BC
AC
(A+B)
(A+C’)
F
G
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
1
1
0
1
1
0
0
1
0
0
1
1
1
1
0
1
1
0
1
0
1
0
1
0
1
1
0
0
0
0
1
1
0
1
1
0
1
1
0
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
b) Show the canonical algebraic expression in sum of products form.
Answer:
F = AB’C’ + A’BC + AB’C + ABC
G = A’BC’ + AB’C’ + ABC’ + AB’C + ABC
c) Show a minimum SOP expression ( : 2 terms, 4 literals; : 2 terms, 3 literals). F G
Answer:
F = BC + AB’
G = A + BC’
d) Show the minterms of the complement of each function in numeric form.
F = ∑m(0,1,2,6)
G = ∑m(0,1,3)
e) Show the canonical algebraic expression in product of sums form.
F = (A + B + C) (A + B + C’) (A + B’ + C) (A’ + B’ + C)
G = (A + B + C) (A + B + C’) (A + B’ + C’)
f) Show a minimum POS expression ( : 2 terms, 4 literals; : 2terms, 4 literals). F G
F = (A + B) (B’ + C)
G = (A + B) (A + C’)
Chapter 2, problem 17: (1 pts) Show that the NOR is functionally complete by implementing a
NOT, a two-input AND, and a two-input OR using only two-input NORs.
Chapter 2, problem 19: (1 pts) Show a block diagram corresponding to each of the expressions
below using only NAND gates. Assume all inputs are available both complemented and
uncomplemented.
c) h = z(x'y + w'x') + w(y' +xz')
Expanding the equation,
h = x’yz + w’x’z + y’w + xz’w
