MAT 214 Discrete Mathematics: Homework 4 questions

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Discrete Mathematics (MAT 214) Homework 4 Complete the following problems on a separate sheet of paper. Write as much as possible for full credit, including calculator work. This assignment is due on Friday, February 22nd . Prove the following statements: 1) If a is any odd integer and b is any even integer, then 2a + 3b is even. 2) If n is any odd integer, then (−1)n = −1. 3) For all integers m, if m > 3 then m2 − 4 is composite. 4) For all integers n and m, if n − m is even then n3 − m3 is even. 5) The following three proofs are connected. The main result is to show that given two rational numbers, you can find another rational number between the two of them. This will be part c. Use the results from a and b to show this. a) If r and s are any two rational numbers, then r+s 2 b) For all real numbers a and b, if a < b then a < ties of inequalities in T17-T27 of Appendix A.) is rational. a+b 2 < b. (You may use the proper- c) Given any two rational numbers r and s with r < s, there is another rational number between r and s. You should use the results from parts a and b to show this.

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School: Duke University

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Discrete Mathematics: Homework 4

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Discrete Mathematics: Homework 4
The attached word document addresses the question “Discrete Mathematics 5 questions” by
Complete the following problems on a separate sheet of paper. Write as much as possible for full
credit, including calculator work. This assignment is due on Friday, February 22nd .
Prove the following statements:
1.

If a is any odd integer and b is any even integer, then 2a + 3b is even.

2.

If n is any odd integer, then (−1)n = −1.

3.

For all integers m, if m > 3 then m2 − 4 is composite.

4. For all integers n and m, if n − m is even then n 3 − m3 is even.
5. The following three proofs are connected. The main result is to show that given two rational

numbers, you can find another rational number between the two of them. This will be part
c. Use the results from a and b to show this.
a. If r and s are any two rational numbers, then r+s 2 is rational.
b. For all real numbers a and b, if a < b then a < (a+b)/2 < b. (You may use the properties

of inequalities in T17-T27 of Appendix A.)
c. Given any two rational numbers r and s with r < s, there is another rational number

between r and s. You should use the results from parts a and b to show this.

1

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