Instructor: Vladimir Filkov

Midterm Review Exercises

1.

Write the truth table for the proposition (r q) (p r). q Ans: p q) (p r) r (r

p

T

T

T

T

F

F

F

F

2.

T

T

F

F

T

T

F

F

T

F

T

F

T

F

T

F

T

T

F

T

T

F

F

F

Find a proposition with three variables p, q, and r that is true when exactly one of the three variables is true, and false otherwise

Ans: (p q r) (p q r) (p q r).

3.

Determine whether p (q r) and p (q r) are equivalent.

Ans: Not equivalent. Let q be false and p and r be true.

4.

Write the contrapositive, converse, and inverse of the following: You sleep late if it is Saturday.

Ans: Contrapositive: If you do not sleep late, then it is not Saturday. Converse: If you sleep late, then it is Saturday. Inverse: If it is not Saturday, then you do not sleep late.

5.

On the island of knights and knaves you encounter two people. A and B. Person A says, "B is a knave." Person B says, "At least one of us is a knight." Determine whether each person is a knight or a knave.

Ans: A is a knave, B is a knight.

In the two questions below P(xy) means “x and y are real numbers such that x 2y 5”.

Determine whether the statement is true.

6.

xyP(xy).

Ans: True, since for every real number x we can find a real number y such that x

2y 5, namely y (5 x)2.

7.

xyP(xy).

Ans: False, if it were true for some number x0, then x0 = 5 -2y for every y, which is not possible.

8.

Determine whether the following argument is valid: pr qr

(p q)

________

r

Ans: Not valid: p false, q false, r true

9.

Prove that the following is true for all positive integers n: n is even if and only if

3n2 8 is even.

Ans: If n is even, then n 2k. Therefore 3n2 8 3(2k)2 8 12k2 8 2(6k2

4), which is even. If n is odd, then n 2k 1. Therefore 3n2 8 3(2k

1)2 8 12k2 12k