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(A. Given S = ZXN Define equivalence relation on 8 #2XN as ca,b)~ Cc,d) if and only if ad-bc = 0. Beflerive, :

(A. Given S = ZXN Define equivalence relation on 8 #2XN as ca,b)~ Cc,d) if and only if ad-bc = 0. Beflerive, :

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 13E: 13. Consider the set of all nonempty subsets of . Determine whether the given relation on is...
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terice the given relation is Equivalence relation. Moreover if a,b are relatively primethe_ad -bc =0 > ad s bc,therefore a divides c and b divides d. Given relation on z defined as ano if and only if 20+b = O mod 0. Refiexiye : clearly a ~a for alla E Z as : 2at9=0a : O mod G Hence he given relation is Reflexive. Symmetric: Let a~o, Therefore 20+b # o mod o. Therefore 20 tb 3Ok for some integer Hence b = OK Now consider: 20 +q = 2COK-20) *a 2a ニ ok 0 mod 9. Hence b ~a, Transitive: Suppose a,b,c e z be such that anb,o~c. Therefore : 2a tb s o mod o and 2b + c 30 mod .
Transcribed Image Text:terice the given relation is Equivalence relation. Moreover if a,b are relatively primethe_ad -bc =0 > ad s bc,therefore a divides c and b divides d. Given relation on z defined as ano if and only if 20+b = O mod 0. Refiexiye : clearly a ~a for alla E Z as : 2at9=0a : O mod G Hence he given relation is Reflexive. Symmetric: Let a~o, Therefore 20+b # o mod o. Therefore 20 tb 3Ok for some integer Hence b = OK Now consider: 20 +q = 2COK-20) *a 2a ニ ok 0 mod 9. Hence b ~a, Transitive: Suppose a,b,c e z be such that anb,o~c. Therefore : 2a tb s o mod o and 2b + c 30 mod .
(A. Given S = ZXN Define equilalence relationon 8 =2XN as Ca,b)~ Cc,d) if and only if ad-bc =O. Reflexive: clearly Ca,b)~Ca,b) because ab - ab =O. Bince Ca,b was an arbitrary element of 8 = Z× N, Therefore every element of the set s = 2XN is related to itself. Hence the giyen relation is Reflexive. Symetric: Let Ca,b), Ca,d)E 8 be such that Ca,b) ~Cc,d) Therefore_ad - bc = O. nad-bu =0 bo-ad =O cb-da =0 Which implies that Ca,d)~(a,b). Hence the gixen relation is Gymmetric . Trancitive: Let ca,b), ce,d) and Ce,f)E 6 be such that: cab)~Ca,d) and Ca,d)~ce, f). Therefore : ad-bc = 0 => ad =bc and cf -de = O => cf =de. adcf #bode bede do (:: C,d EN) adcf do af =be Which implies that ca,b)~Ce,f). Hence the given relation is Transitive.
Transcribed Image Text:(A. Given S = ZXN Define equilalence relationon 8 =2XN as Ca,b)~ Cc,d) if and only if ad-bc =O. Reflexive: clearly Ca,b)~Ca,b) because ab - ab =O. Bince Ca,b was an arbitrary element of 8 = Z× N, Therefore every element of the set s = 2XN is related to itself. Hence the giyen relation is Reflexive. Symetric: Let Ca,b), Ca,d)E 8 be such that Ca,b) ~Cc,d) Therefore_ad - bc = O. nad-bu =0 bo-ad =O cb-da =0 Which implies that Ca,d)~(a,b). Hence the gixen relation is Gymmetric . Trancitive: Let ca,b), ce,d) and Ce,f)E 6 be such that: cab)~Ca,d) and Ca,d)~ce, f). Therefore : ad-bc = 0 => ad =bc and cf -de = O => cf =de. adcf #bode bede do (:: C,d EN) adcf do af =be Which implies that ca,b)~Ce,f). Hence the given relation is Transitive.
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