Discrete structures in mathematics and computer science

Q1) Use the standard logical equivalences to simplify the expression
(ㄱp ^ q) v ㄱ(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, 3ㅣn^3+2n (Hint, consider 3 separate cases)

Q5) For the following sets A and B find A∪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}.