1. Let A = {1, 2} and B = {x, y}. (a) Find (AXA) XB. (b) Find AxAxB. 2. Let S = {A, B, C). List all strings of length 3 over S that have at least two B's. 3. Let A = {2, 3, 4} and B = {4, 5, 6} and define relations R and S from A to B as follows: For every (x, y ) EAXB, (x,y) ER means that -x2 - is an integer (x, y) ES means that y = 8-x 3 (a) Draw arrow diagrams for R and S. (b) Indicate whether either R or S is a function from A to B. 4. Find all functions from {1, 2} to {s, t}. In problems 5 and 6, use truth tables to determine if the following statements are logically equivalent. 5. pa(q-r) and ( par) v(pa~q) 6. (p+q)-r and ~(paq)vr 7. Use Theorem 2.1.1 to verify the logical equivalence. Supply a reason for each step. ~(pvq ) v ( ~ paq ) = ~p 8. Write the contrapositive, converse, and inverse of the following conditional statement. If you do not study, then you will struggle in this class. 9. Rewrite each of the following statements in if-then form. a) To pass the course it is necessary to write an essay. (b) A sufficient condition for being happy is eating ice cream. 10. Write negations for each of the following statements. (a) If you don't eat your meat, then you can't have any pudding. (b) If n is divisible by 10, then n is divisible by 2 and n is divisible by 5