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6, Suppose a,b,c cZ. If a b and a c, then a (b+ e). 7 Suppose a,be Zz. If a | b, then a |62. 8. Suppose a is an integer. If5 | 2a, then 5 a. 9. Suppose a is an integer. If 7 | 4a, then 7 |a.

6, Suppose a,b,c cZ. If a b and a c, then a (b+ e). 7 Suppose a,be Zz. If a | b, then a |62. 8. Suppose a is an integer. If5 | 2a, then 5 a. 9. Suppose a is an integer. If 7 | 4a, then 7 |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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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 = 20 +26+1 = 2(a +b)+1, which is odd (by Definition 42, Proposition If two integers have opposite parity, then their sum is 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 section in 126 Direct Pr lar Case 請调游戏名词 ger and and e is We need to show that m +n is odd some integers a and h 心事 n = 2( ) is 2 2, t Thus m = 2a and n= 26 + 1 hus fo here 电影院 a the les spirit, egen 态度 EN. Su to exercises. The the back of the text. oda de dgeno 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 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,d e Z. If a |b and c|d, then ac bd. 12. If xeR 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 ne Z, then 5n² + 3n + 7 is odd. (Try cases.) g ad o (15, If n € Z, then n² +3n + 4 is even. (Try cases.) net bho bna nas sd (16. If two integers have the same parity, then their sum is even. (Try cases.) 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 e are integers. If a² | b and b³ | c, then aº | c. 爱好
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 = 20 +26+1 = 2(a +b)+1, which is odd (by Definition 42, Proposition If two integers have opposite parity, then their sum is 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 section in 126 Direct Pr lar Case 請调游戏名词 ger and and e is We need to show that m +n is odd some integers a and h 心事 n = 2( ) is 2 2, t Thus m = 2a and n= 26 + 1 hus fo here 电影院 a the les spirit, egen 态度 EN. Su to exercises. The the back of the text. oda de dgeno 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 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,d e Z. If a |b and c|d, then ac bd. 12. If xeR 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 ne Z, then 5n² + 3n + 7 is odd. (Try cases.) g ad o (15, If n € Z, then n² +3n + 4 is even. (Try cases.) net bho bna nas sd (16. If two integers have the same parity, then their sum is even. (Try cases.) 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 e are integers. If a² | b and b³ | c, then aº | c. 爱好
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