Question

Do not use the same proof approach more than once (use direct, contr contradiction). Justify your choice of proof approach for each theory. If n is an even integer, then -3n - 1 is an odd integer. If 5x - 7 is odd, then 9x + 2 is even. 11x - 7 is even if and only if x is odd.

Answer 1

To prove that n is an even integer implies -3n - 1 is an odd integer, assume n is even and substitute it into the given expression. To prove that 5x - 7 is odd implies 9x + 2 is even, assume 5x - 7 is odd and substitute it into the given expression. To prove that 11x - 7 is even if and only if x is odd, assume 11x - 7 is even and substitute it into x. Then assume x is odd and substitute it into 11x - 7.

Step-by-step explanation:

To prove that n is an even integer implies -3n - 1 is an odd integer:

1. Assume that n is an even integer.
2. Express -3n - 1 as -3(n) - 1.
3. Since n is even, it can be written as 2k, where k is an integer.
4. Substitute 2k for n in -3(n) - 1.
5. Simplify the expression to obtain -6k - 1.
6. Since k is an integer, -6k is always an even integer.
7. The sum of an even and an odd integer is always an odd integer.
8. Therefore, -6k - 1 is an odd integer.
9. This proves that if n is an even integer, then -3n - 1 is an odd integer.

To prove that 5x - 7 is odd implies 9x + 2 is even:

1. Assume that 5x - 7 is odd.
2. Express 5x - 7 as 5(x) - 7.
3. Since 5x - 7 is odd, it can be written as 2k + 1, where k is an integer.
4. Substitute 2k + 1 for 5x - 7 in 9x + 2.
5. Simplify the expression to obtain 9(x) + 2.
6. Expand 9(x) to obtain 9x.
7. Combine like terms to obtain 9x + 2.
8. Since 9x is always an even integer and 2 is an even integer, their sum is always an even integer.
9. Therefore, 9x + 2 is an even integer.
10. This proves that if 5x - 7 is odd, then 9x + 2 is even.

To prove that 11x - 7 is even if and only if x is odd:

1. Assume that 11x - 7 is even.
2. Express 11x - 7 as 11(x) - 7.
3. Since 11x - 7 is even, it can be written as 2k, where k is an integer.
4. Substitute 2k for 11x - 7 in x.
5. Simplify the expression to obtain 2(x) - 7.
6. Expand 2(x) to obtain 2x.
7. Combine like terms to obtain 2x - 7.
8. Since 2x is always an even integer and 7 is an odd integer, their difference is always an odd integer.
9. Therefore, 2x - 7 is an odd integer.
10. This proves that if 11x - 7 is even, then x is odd.
11. To prove that x is odd implies 11x - 7 is even:
12. Assume that x is odd.
13. Express x as 2k + 1, where k is an integer.
14. Substitute 2k + 1 for x in 11x - 7.
15. Simplify the expression to obtain 11(2k + 1) - 7.
16. Expand 11(2k + 1) to obtain 22k + 11.
17. Combine like terms to obtain 22k + 11 - 7.
18. Simplify further to obtain 22k + 4.
19. Factor out 2 from 22k + 4 to obtain 2(11k + 2).
20. Since 11k + 2 is always an even integer, 2(11k + 2) is also even.
21. Therefore, 11x - 7 is even if x is odd.