I need a help to answer some questions in Discrete Mathmatics

I need a help to answer some questions in Discrete Mathmatics

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 even

2. 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 Z

9. Prove ∪( − )=( ∪ )−( ∪ ) for all sets X, Y and Z

10. Prove − ̅̅̅̅̅̅̅̅= ∪ ̅ for all sets X and Y

11. Prove −( ∪ )=( − )∪ for all sets X, Y and Z

12. 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 ≥1

16. 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

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