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Prove the following statements. Use only definitions from Chapter 4 and the assumptions shown below. 1. For all integers a, b, and c, if a | b and a | c then a | (2b - 3c). 2. If a is any odd integer, then a^2 + a is even. 3. For any rational numbers r and s, 2r + 3s is rational. Assumptions: In this text, we assume a familiarity with the laws of basic algebra, listed in Appendix A. We also use the three properties of equality: For all objects A, B, and C, i) A = A ii) if A = B then B = A, and iii) if A = B and B = C, then A = C. We also use the substitution property – for all objects A and B, if A = B, then we may substitute B wherever we have A. 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.

          Prove the following statements. Use only definitions from Chapter 4 and the assumptions shown below.

1. For all integers a, b, and c, if a | b and a | c then a | (2b - 3c).
2. If a is any odd integer, then a^2 + a is even.
3. For any rational numbers r and s, 2r + 3s is rational.

Assumptions:
In this text, we assume a familiarity with the laws of basic algebra, listed in Appendix A.
We also use the three properties of equality: For all objects A, B, and C,
i) A = A
ii) if A = B then B = A, and
iii) if A = B and B = C, then A = C.
We also use the substitution property – for all objects A and B, if A = B, then we may substitute B wherever we have A.
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.

prove the following statements use only definitions from chapter 4 and the assumptions shown on the next page 1 for all integers a b and c if a b and a c then a 2b 3c 2 if a is any odd integ 94963

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