i need a help to answer some questions in discrete mathmatics 1

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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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