Question

show that the relation x≡y(mod 3) defined on the set of integers Z is an equivalence relation

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Answer 1

Step-by-step explanation:

Let a be any number in Z,

Then, a-a=0 is an even integer .

So, Every number is related to itself .

Thus the relation is reflexive .

Let a,b be two numbers in Z,

Also let a−b=k be an even integer (which implies aRb)

Then, b−a=−k is also an even integer (which implies bRa).

Thus, It is evident aRb↔bRA

Thus the relation is also symmetric .

Let a,b,c be three numbers in Z,

Also let aRb and bRc,

then a−b=2k

1

and b−c=2k

2

for k

1

,k

2

belonging to Z

which gives a−b+b−c=2k

1

+2k

2

⇒a−c=2(k

1

+k

2

)

which implies aRc .

Thus relation is also transitive .

Thus we finally arrive at a conclusion that relation is Equivalence relation .

Thanku so much

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