See the attached file for questioins

1.               Prove the following statement:
For all integers n, if n is odd then is odd.

2. Use mathematical induction to show that
1 3 + 2 3 + 3 3 + ... + n 3 = n 2 (n + 1) 2 / 4

3. Prove that the sum of an even integer and an odd integer is always odd.

4. Let B(x) be “x is a bird”, F(x) be “x has feathers”, and Y(x) be “x can fly”.

5. Negate the following quantified statements:

All dogs are loyal.

Some movies are over three hours long.

6. Prove the following property using a set-membership table:

For all sets A, B and C, (A-B)∪(B-C)=(A∪B)-(B∩C)

7. Prove using any valid method that for all sets A and B, (A-B)∩(A∩B)=ϕ

8. Disprove the following statement by giving a counterexample:
For all integers n, if n is odd then is odd.