Prove the statement. In each case use only the definitions of the terms and the Assumptions, not any previously established properties of odd and even integers. Follow the directions given in this section for writing proofs of universal statements.
Definition
An integer n is even if, and only if, n equals twice some integer. An integer n is odd if, and only if, n equals twice some integer plus 1.
Symbolically, if n is an integer, then
n is even ⇔ ∃ an integer k such that n = 2k.
n is odd ⇔ ∃ an integer k such that n = 2k + 1.
Assumptions
• In this text we assume a familiarity with the laws of basic algebra, which are listed in Appendix A.
• We also use the three properties of equality: For all objects A, B, and C,
(1) A = A, (2) if A = B then B = A, and (3) if A = B and B = C, then A = C.
• In addition, we assume that there is no integer between 0 and 1 and that the set of all integers is closed under addition, subtraction, and multiplication. This means that sums, differences, and products of integers are integers.
• Of course, most quotients of integers are not integers. For example, 3 ÷ 2, which equals 3/2, is not an integer, and 3 ÷ 0 is not even a number.
Exercise
The negative of any even integer is even.
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