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if Z is the set of all integers and R is the relation on Z defined as R ={(a, b) :a b € z and a-b is divisible by 5 } prove that x is an equivalence relation
Asked by aiswaryaanil0009 | 01 Feb, 2024, 08:36: PM

Relation R is given as R ={(a, b) :a b € z and a-b is divisible by 5 }

Reflexive:

For any integer a, we have a - a = 0, which is divisible by 5.

So, aRa.

Symmetric:

Let aRb i.e. (a - b) is divisible by 5.

i.e. a - b = 5k for some integer k

Therefore, b - a = 5m for some integer m = -k

So, (b - a) is divisible by 5.

Therefore, bRa.

Transitive:

Let aRb and bRc

Therefore, (a - b) and (b - c) are divisible by 5.

Therefore, (a - b) = 5k & (b - c) = 5m for integers k & m.

Now, (a - c) = (a - b) + (b - c) = 5k - 5m = 5(k - m).

Here, (k - m) is also an integer.

So, (a - c) is divisible by 5.

Therefore, aRc.

So, R is reflexive, symmetric and transitive.

Hence, T is an equivalence relation.

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