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H 25. For all integers n, if n² is odd then n is odd. 26 For all integers a h and if a

H 25. For all integers n, if n² is odd then n is odd. 26 For all integers a h and if a

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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How do you write the contradiction for the proof of 25?
oducts (h) nator, refore ich is state- irra- hal, a iffer- mber of in 1 in and irra- onal east need to show to prove b. Write what you would suppose and what you would need to show to prove this statement by contraposition. 22. Consider the statement "For all real numbers r, if r² is irra- tional then r is irrational." a. Write what you would suppose and what you would need to show to prove this statement by contradiction. b. Write what you would suppose and what you would need to show to prove this statement by contraposition. Prove each of the statements in 23-29 in two ways: (a) by con- traposition and (b) by contradiction. 23. The negative of any irrational number is irrational. 24. The reciprocal of any irrational number is irrational. (The reciprocal of a nonzero real number x is 1/x.) H 25. For all integers n, if n² is odd then n is odd. 26. For all integers a, b, and c, if a X bc then a Xb. (Recall that the symbol X means "does not divide.") H 27. For all integers m and n, if m +n is even then m and n are both even or m and n are both odd. 28. For all integers m and n, if mn is even then m is even or n is even. 0011 29. For all integers a, b, and c, if a | b and a Xc, thening a X (b + c). (Hint: To pro P→q V r, it suffices to prove either p^~qror p^rq. See Section 2.2.) 14 in
Transcribed Image Text:oducts (h) nator, refore ich is state- irra- hal, a iffer- mber of in 1 in and irra- onal east need to show to prove b. Write what you would suppose and what you would need to show to prove this statement by contraposition. 22. Consider the statement "For all real numbers r, if r² is irra- tional then r is irrational." a. Write what you would suppose and what you would need to show to prove this statement by contradiction. b. Write what you would suppose and what you would need to show to prove this statement by contraposition. Prove each of the statements in 23-29 in two ways: (a) by con- traposition and (b) by contradiction. 23. The negative of any irrational number is irrational. 24. The reciprocal of any irrational number is irrational. (The reciprocal of a nonzero real number x is 1/x.) H 25. For all integers n, if n² is odd then n is odd. 26. For all integers a, b, and c, if a X bc then a Xb. (Recall that the symbol X means "does not divide.") H 27. For all integers m and n, if m +n is even then m and n are both even or m and n are both odd. 28. For all integers m and n, if mn is even then m is even or n is even. 0011 29. For all integers a, b, and c, if a | b and a Xc, thening a X (b + c). (Hint: To pro P→q V r, it suffices to prove either p^~qror p^rq. See Section 2.2.) 14 in
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