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nd 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 mar

nd 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 mar

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 proof of number 25?
se products y the (h) nominator, e therefore 1, which is the state- any irra- ational, a he differ- number proof in se 11 in ber and y irra- tional least n is odd." a. Write what you would suppose and what you w b. Write what you would suppose and what you wou need to show to prove this statement by contradiction need to show to prove this statement by contrapositi 22. Consider the statement "For all real numbers r, if r² is i tional then r is irrational." a. Write what you would suppose and what you wo b. Write what you would suppose and what you wo need to show to prove this statement by contradiction need to show to prove this statement by contraposition Prove each of the statements in 23-29 in two ways: (a) by ca 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 is even. 29. For all integers a, b, and c, if a | b and a Xc, then a X (b+c). (Hint: To prove pqVr, it suffices to prove either p^~qror prq. See exercise 14 in Section 2.2.) its squ then 32. U in a An 1. th then
Transcribed Image Text:se products y the (h) nominator, e therefore 1, which is the state- any irra- ational, a he differ- number proof in se 11 in ber and y irra- tional least n is odd." a. Write what you would suppose and what you w b. Write what you would suppose and what you wou need to show to prove this statement by contradiction need to show to prove this statement by contrapositi 22. Consider the statement "For all real numbers r, if r² is i tional then r is irrational." a. Write what you would suppose and what you wo b. Write what you would suppose and what you wo need to show to prove this statement by contradiction need to show to prove this statement by contraposition Prove each of the statements in 23-29 in two ways: (a) by ca 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 is even. 29. For all integers a, b, and c, if a | b and a Xc, then a X (b+c). (Hint: To prove pqVr, it suffices to prove either p^~qror prq. See exercise 14 in Section 2.2.) its squ then 32. U in a An 1. th then
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