1. Use the standard logical equivalences to simplify the expression (12593; p ^ q) v 12593;(pVq)
2. Consider the following theorem "The square of every odd natural number is again an odd number" What is the hypothesis of the theorem? what is the conclusion? give a direct proof of the theorem.
3. Consider the following theorem "The sum of a rational number and an irrational number is an irrational number. What is the hypothesis of the theorem? what is the conclusion? Give a direct proof of the theorem.
4. Prove that for any integer n, 312643;n^3+2n (Hint, consider 3 separate cases)
5. For the following sets A and B find A8746;B, A8745;B and AB.
a) A={1,2,a} B={2,3,a}
b) A={2,7,b), B={7,3,4}
c) A=Z, B=N
6. Write down the power sets for each of the following sets:
a) 966;
b) {966;}
c) {4,7}
7. Find the Cartesian products A*B, B^2 and A^3 for the sets A={0,x} and B={0,1,4}.