Question

show that the relation R in the set Z of integers given by R={(x,y)/2 divides x-y} is an equivalence relation
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Answer 1

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Answer 2

Step-by-step explanation:

2{(a,b)=2dividesa=b}

where R is in set 2 of integers

a-a=0

2 divides a-a

⇒(a,a)ϵR ∴R is reflexive

Let (a,a)ϵR

∴2 divides a−b⇒a−b=2n

for same nϵz⇒b−a=2(−n)

⇒ 2 divides b−a=(b,a)ϵR

R is symmetric

Let (a,b) & (b,c)ϵR

2 divides a-b & b-c

∴a−b=2n & b−c=2n

2

for same n

1

,n

2

ϵ2

=2n

1

+2n

2

=a−c=2(n

1

+n

2

)

∴2 divides a-c

(a,c)ϵR

∴(a,b)(b,c)ϵR⇒(a,c)ϵR

∴R is transitive

So, R is an equivalence relation