1. (5 points) Find a universal existential statement which formalizes the following statement: Every family in the U.S. has a SUV.

2. (10 points) Use both set-roster and set-builder notations to specify the following sets:

(a) A set of positive real numbers that can be written as the square root of an integer.

(b) Set of all negative odd integers that are not less than -7.

3. Let A = {10, 10, 9, 11, 8}, B = {n Z|8 n < 12} and C = { 64, 10, 3 2 , 11, 33 3 }.

(a) (5 points) Is A B? Why?

(b) (5 points) Is B A? Why?

(c) (5 points) Is C B? Why?

(d) (10 points) Specify the set A B using both set-roster and set-builder notations.

(e) (15 points) Let R denote a relation from A to B such that: (x, y) R means that (y x) is not a prime number. Specify all the ordered pairs which belong to R.

(f) (10 points) Does R specify a function? Why?

4. Given statement variables p and q, verify the following logical equivalences by constructing a truth table for each part:

(a) (10 points) p (c p) p t

(b) (10 points) (p q) (p q) ( p q)

5. Given any statement variables p, q, and r,

(a) (10 points) Is p (q r) logically equivalent to (p q) ( p r)?

(b) (5 points) Is t (c t) c (c t)? Prove your answer by constructing the truth table of each statement form.