Determine whether the statement is true or false. Justify your answer with a proof or a counterexample, as appropriate. In each case use only the definitions of the terms and the Assumptions not any previously established properties.
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
For all nonnegative real numbers a and b, (Note that if x is a nonnegative real number, then there is a unique nonnegative real number y, denoted such that y2 = x.)
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