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Use the method of contrapositive proof to prove the following statements. (In each case you should also think about how a direct proof would work. You will find in most cases that contrapositive is easier.) 1. Suppose ne Z. If n² is even, then n is even. 2. Suppose ne Z. If n² is odd, then n is odd. 3. Suppose a,b € Z. If a²(b²-2b) is odd, then a and b are odd. 4. Suppose a,b,c e Z. If a does not divide be, then a does not divide b. 5. Suppose x € R. If x² +5x<0 then x<0. D 16.3

Use the method of contrapositive proof to prove the following statements. (In each case you should also think about how a direct proof would work. You will find in most cases that contrapositive is easier.) 1. Suppose ne Z. If n² is even, then n is even. 2. Suppose ne Z. If n² is odd, then n is odd. 3. Suppose a,b € Z. If a²(b²-2b) is odd, then a and b are odd. 4. Suppose a,b,c e Z. If a does not divide be, then a does not divide b. 5. Suppose x € R. If x² +5x<0 then x<0. D 16.3

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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Use the method of contrapositive proof to prove the following statements. (In each case you should also think about how a direct proof would work. You will find in most cases that contrapositive is easier.) 1. Suppose n € Z. If n² is even, then n is even. 2. Suppose n e Z. If n² is odd, then n is odd. 3. Suppose a,b € Z. If a²(b²-2b) is odd, then a and b are odd. 4. Suppose a,b,c € Z. If a does not divide be, then a does not divide b. 5. Suppose x € R. If x² +5x<0 then x < 0. 6. Suppose x E R. If x-x>0 then x>-1. 7. Suppose a, b € Z. If both ab and a+b are even, then both a and b are even. 8. Suppose x € R. If x5-4x²+3x³-x²+3x-4≥0, then x≥0. 9. Suppose n € Z. If 3+n², then 3 n. 10. Suppose x,y,ze Z and x 0. If xyz, then xy and xtz.
Transcribed Image Text:Use the method of contrapositive proof to prove the following statements. (In each case you should also think about how a direct proof would work. You will find in most cases that contrapositive is easier.) 1. Suppose n € Z. If n² is even, then n is even. 2. Suppose n e Z. If n² is odd, then n is odd. 3. Suppose a,b € Z. If a²(b²-2b) is odd, then a and b are odd. 4. Suppose a,b,c € Z. If a does not divide be, then a does not divide b. 5. Suppose x € R. If x² +5x<0 then x < 0. 6. Suppose x E R. If x-x>0 then x>-1. 7. Suppose a, b € Z. If both ab and a+b are even, then both a and b are even. 8. Suppose x € R. If x5-4x²+3x³-x²+3x-4≥0, then x≥0. 9. Suppose n € Z. If 3+n², then 3 n. 10. Suppose x,y,ze Z and x 0. If xyz, then xy and xtz.
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