Assignment:
Q1) Use the standard logical equivalences to simplify the expression
(¬p ^ q) v ¬(pVq)
Q2) 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.
Q3) 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
Q4) Prove that for any integer n, 3|n^3+2n (Hint, consider 3 separate cases)
Q5) For the following sets A and B find A U B, A ∩ 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
Q6) Write down the power sets for each of the following sets:
a) φ b) {φ} c) {4,7}
Q7) Find the Cartesian products A*B, B^2 and A^3 for the sets A={0,x} and B={0,1,4}.
Provide complete and step by step solution for the question and show calculations and use formulas.