1). Insert parentheses in the following formula

p q r

to identify the proper procedure for evaluating its truth value. Construct its truth table

2).Insert parentheses in the following formula

p q p q.

to identify the proper procedure for evaluating its truth value. Construct its truth table

3). Use a truth table to show that

[(p q) r] [r (p q)]

4). The logical connective exclusive or, denoted p Y q, means either p

or q but not both. Consequently,

p Y q (p q) (p q) (p q) (p q).

Construct a truth table to verify this claim

5).The proposition

"The square of any real number is positive"

is a universal quantification

"For any real number x, x

2 > 0."

Is it true or false?

6), Show that 1

3

a +

2

3

b is closer to b than to a.

Hint: Compute the distance between a and 1

3

a +

2

3

b, and compare it to the distance between

1

3

a +

2

3

b and b.

7). Show that, between any two rational numbers a and b, where a < b,

there exists another rational number closer to b than to a.

Hint: Use a weighted average of a and b.

8).Let n be an integer. Show that if n is odd, then n

3

is odd

9).Show that n

3 + n is even for all n N

10).Let x be a real number. Prove that if (2x

2 + 3)(x + 5)(x 7) = 0,

then either x = 5, or x = 7.

11).Use logical equivalencies to show the two statements are equivalent:

Statement 1: (P Q) (Q P)

Statement 2: P Q

Hint: Start with Statement 1. Remember, P Q is equivalent to P Q. Use the Absorption Rule or

Associative Rule as necessary.