Complete the following homework by the due date. All work submitted must be your own. When applicable show your work. Remember, Prove may mean prove or disprove.1. Prove if a and b are integers, prove that if a is even and b is even then a+b is even2. Prove ∀ ∈ ∀ ∈ ( →( × ) )3. Prove ∀ ∈ ∀ ∈ ( ( × )∈ )4. Prove ∀ ∈ ( ≠0→(1 )∈ )5. Prove if m is a perfect square and n is a perfect square, then × is a perfect square.6. Prove that for an integer , 2− is divisible by 2.7. Prove ⊆ ∪ , for all sets , 8. Prove ∩( − )=( ∩ )−( ∩ ) for all sets X, Y and Z9. Prove ∪( − )=( ∪ )−( ∪ ) for all sets X, Y and Z10. Prove − ̅̅̅̅̅̅̅̅= ∪ ̅ for all sets X and Y11. Prove −( ∪ )=( − )∪ for all sets X, Y and Z12. Using a proof by contradiction prove that ∀ ∈ℛ 2 ℎ 13. Using a proof by contradiction prove ∀ ∈ ( 2 ℎ )14. Using a proof by contrapositive prove ∀ ∈ ( 2 ℎ )15. Prove for all real number x, y and z. If + + ≥3 ℎ ≥1 ≥1 ≥116. Prove that 3+2 2=36 has no solution in positive integers.17. Use a proof by cases to prove that | + |=| |+| | for all real numbers, x and y