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hey steve its eva can you help me on homework #theoem24-28

exlpain, show work and verify your answer is correct, use truth table if needed, show defintion if needed,or used,

Deﬁnition 12. A statement of the form A and B (sometimes written A^B) is true exactly when A is true and B is true. A statement of the form A or B (sometimes written A\/B) is false exactly when A is false and B is false. A statement of the form A implies B, also known as “If A, then B,” and also written A => B, is false exactly when A is true and B is false. A statement of the form “not A” (sometimes written¬A, and also called the negation of A) is true exactly when A is false.

Deﬁnition 13. A statement of the form A if and only if B, sometimes written A iff B, or A<=>B or A is equivalent to B, means A implies B and B implies A.

Deﬁnition 16. The contrapositive of a statement of the form A => B is the statement¬B => ¬A.

Theorem 24. For all integers n, x and y, if n divides x, and n divides y, then n divides x+y.

Theorem 25. For all integers n, x, and y, if n divides x, and n divides y, then n divides(x-y).

Prove each of Propositions 26 and 27 using the contrapositive of the given statement:

Theorem 26. Assume x,y are integers. If xy is odd, then both x and y are odd.

Theorem 27. Assume x,y are integers. If xy is even, then x or y is even.

Theorem 28. Assume x and y are integers. Then xy is odd if and only if x and y are odd. (First, use the deﬁnition of equivalence to separate the “if and only if” into two separate statements. Then, prove each statement.)

Deﬁnition 1. An integer n is even if n=2k for some integer k.

Deﬁnition 2. An integer n is odd if n=2k+1 for some integer k.

Deﬁnition 5. An integer n divides the integer m, written n|m, if there exists another integer k such that m=nk. In the same context, we may also write that m is divisible by n.

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