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15, If neZ, then nº +3n +4 is even. (Try cases.) ho bn 6. If two integono

15, If neZ, then nº +3n +4 is even. (Try cases.) ho bn 6. If two integono

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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of transparency we will avoid writing it this way. In a similar Use the method of direct proof to prove the following statements, Proof. Suppose m and n are two integers with opposite parity. Therefore m +n = 2a +26+1= 2(a +b)+1, which is odd (by Definition A Proposition If two integers have opposite parity, then their wam In reading proofs in other texts, you may sometimes see the phr Please check your understanding by doing the following e "Without loss of generality" abbreviated as "WLOG." However, in the intere advisable-at least until you become more experienced in proof writ that you write out all cases, no matter how similar they appear to be odd numbered problems have complete proofs in the Solutions secetion in 126 Dreet Pr lar Case 請調游戏名词 ger and and e im We need to show that m +n is odd. some integers a and h 心事 n = 2(: ) is 2 2, t hus fa here Thus m= 2a and n= 26 + 1 电影院 a the nd e des spirit, egen 态度 EN. Sup exercises. The the back of the text. roda oode 钥匙 Exercises for Chapter 4 1. If x is an even integer, then x is even. 2. If x is an odd integer, then r is odd. 3. Ifa is an odd integer, then a +3a +5 is odd. 4 Suppose x, y € Z. If x and y are odd, then xy is odd. 5. Suppose x,yEZ. Ifx is even, then xy is even. 6 Suppose a, b,ce Z. If a b and a c, then a | (b+ c). 7 Suppose a, be Z. If a b, then a² | 62. 8. Suppose a is an integer. If 5 | 2a, then 5|a. 9. Suppose a is an integer. If 7 | 4a, then 7|la. 10. Suppose a and b are integers. If a |b, then a | (363 -62 +5b) 11. Suppose a, b,c,de Z. If a b and e d, then ac bd. 12. If reR and 0<x< 4, then 21. 13. Suppose x,y €R. If x² +5y = y² +5x, then x = y or x+y=5. 14. If n €Z, then 5n2 + 3n + 7 is odd. (Try cases.) ad s o (15, If n €Z, then n² +3n +4 is even. (Try cases.) net bho boa na s (16. If two integers have the same parity, then their sum is even. (Try cases.) 爱好 17. If two integers have opposite parity, then their product is even. 18. Suppose x and y are positive real numbers. If x <y, then x² < y². 19. Suppose a,b and c are integers. If a² |b and b³ | c, then aº | c.
Transcribed Image Text:of transparency we will avoid writing it this way. In a similar Use the method of direct proof to prove the following statements, Proof. Suppose m and n are two integers with opposite parity. Therefore m +n = 2a +26+1= 2(a +b)+1, which is odd (by Definition A Proposition If two integers have opposite parity, then their wam In reading proofs in other texts, you may sometimes see the phr Please check your understanding by doing the following e "Without loss of generality" abbreviated as "WLOG." However, in the intere advisable-at least until you become more experienced in proof writ that you write out all cases, no matter how similar they appear to be odd numbered problems have complete proofs in the Solutions secetion in 126 Dreet Pr lar Case 請調游戏名词 ger and and e im We need to show that m +n is odd. some integers a and h 心事 n = 2(: ) is 2 2, t hus fa here Thus m= 2a and n= 26 + 1 电影院 a the nd e des spirit, egen 态度 EN. Sup exercises. The the back of the text. roda oode 钥匙 Exercises for Chapter 4 1. If x is an even integer, then x is even. 2. If x is an odd integer, then r is odd. 3. Ifa is an odd integer, then a +3a +5 is odd. 4 Suppose x, y € Z. If x and y are odd, then xy is odd. 5. Suppose x,yEZ. Ifx is even, then xy is even. 6 Suppose a, b,ce Z. If a b and a c, then a | (b+ c). 7 Suppose a, be Z. If a b, then a² | 62. 8. Suppose a is an integer. If 5 | 2a, then 5|a. 9. Suppose a is an integer. If 7 | 4a, then 7|la. 10. Suppose a and b are integers. If a |b, then a | (363 -62 +5b) 11. Suppose a, b,c,de Z. If a b and e d, then ac bd. 12. If reR and 0<x< 4, then 21. 13. Suppose x,y €R. If x² +5y = y² +5x, then x = y or x+y=5. 14. If n €Z, then 5n2 + 3n + 7 is odd. (Try cases.) ad s o (15, If n €Z, then n² +3n +4 is even. (Try cases.) net bho boa na s (16. If two integers have the same parity, then their sum is even. (Try cases.) 爱好 17. If two integers have opposite parity, then their product is even. 18. Suppose x and y are positive real numbers. If x <y, then x² < y². 19. Suppose a,b and c are integers. If a² |b and b³ | c, then aº | c.
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