prove the statements in 1–11. In each case use only the definitions of the terms and the assumptions listed on page 161, not any previously established properties of odd and even integers. Follow the directions given in this section for writing proofs of universal statements. For every integer n, if n is odd then 3n15 is even For every integer n, 2n21 is odd. For any integers m and n, if m is even and n is odd then 5m13n is odd. If n is any even integer, then (21) n 5 1. pg 161 Assumptions ●● In this text we assum
prove the statements in 1–11. In each case use only the definitions of the terms and the assumptions listed on page 161, not any previously established properties of odd and even integers. Follow the directions given in this section for writing proofs of universal statements.
For every integer n, if n is odd then 3n15 is even
For every integer n, 2n21 is odd.
For any integers m and n, if m is even and n is odd then 5m13n is odd.
If n is any even integer, then (21) n 5 1.
pg 161 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 5 A, (2) if A 5 B, then B 5 A, and (3) if A 5 B and B 5 C, then A 5 C. ●● And we use the principle of substitution: For all objects A and B, if A 5 B, then we may substitute B wherever we have A. ●● 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.
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