A+ Answers

1. Determine whether the compound proposition below is a tautology, contingency or contradiction.

(¬ p ∧ (p → q)) → ¬ q

2.

 Let A = {a, b, c, d, e},              

B = {a, b, c, d, e, f, g, h}.

Find the following sets:

a)     B – A

b)     A ∩ B

 c)     A – B

d)      A ∪ B

3.

Suppose g : A → B and f : B → C are functions where

A = {2, 4, 6, 8}, B = {w, x, y, z}, C = {-1, 0, 1};

g is defined by g = {(2, x), (4, w), (6, y), (8, w), (10, z)}; and

f is defined by f = {(w, 0), (x, 0), (y, 1), (z, -1)}.

Find f ◦ g.

a)     {(2, -1), (4, 0), (6, 0), (8, 1). (10, 1)}

b)    {(2, 0), (4, 0), (6, 1), (8, 0), (10, -1)}

c)     {(-1, z), (z, 10), (0, w), (w, 8), (1, y), (y, 6)}

d)      {(-1, -10), (0, 8), (1 6), (0, 4), (-1, 2)}

4.

Let matrix C be the meet of A and B, that is, .

Which of the following statements is true?

a)     A is symmetric

b)     B is symmetric

c)     C is symmetric

5.

List all the ordered pairs in the relation

R = {(a, b) | a divides b}

on the set {1, 2, 3, 4, 5, 6}.

a)     (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)

b)     (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 2) (2, 4) (2, 6)

c)     (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 2) (2, 4) (2, 6) (3, 3) (3, 6)

d)     (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 2) (2, 4) (2, 6) (3, 3) (3, 6) (4, 4) (5, 5)

e)    (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 2) (2, 4) (2, 6) (3, 3) (3, 6) (4, 4) (5, 5) (6, 6)

6.

Give a formula for a_n if the sequence {a_n} has

a₁ = 19, a₂ = 11, a₃ = 3, a₄ = – 5, a₅ = – 13.

a)      19 – 8n

b)     27 – 8n

c)       1 + 31n– 13n²

d)      15 + (– 1)ⁿ⁺¹(4)

7.

How many edges does an undirected graph have if it has vertices of degree 5, 4, 4, 3, 3, 2, 2, and 1?

8.

What kind of graph is shown here?

a)      K_3,3

b)      K₆

c)      C₆

d)      W₆

e)     K_2,4

f)       None of the above

9.

For the tree shown below, list of vertices of the descriptions a-d:

a)      ancestors of vertex m

b)      children of vertex b

c)      siblings of vertex d

d)      leaves

Example answer would be: e, f, g, h , i, j

            a) The ancestors of vertex m

            b) The children of vertex b

            c) The siblings of vertex d

            d) The leaves