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  • Let n be a fixed positive integer. Define a relation R in Z as follows: ∀ a, b ∈ Z, aRb if and only if a – b is divisible by n. Show that R is an equivalence relation.

Let n be a fixed positive integer. Define a relation R in Z as follows:   \forall a, b \in Z, aRb if and only if a - b is divisible by n. Show that R is an equivalence relation.

Answers (1)

Here, we have to a relation R in Z as follows:   \forall a, b \in Z, aRb if and only if a - b is divisible by n.

Here, aRa   \Rightarrow (a - a) is divisible by n and this is true for all integers.

Therefore, R is reflective.

For aRb, aRb \Rightarrow (a - b)  is also divisible by n.

or, - (b - a) is divisible by n.

or, (b - a) is divisible by n

Hence, we can write it as bRa.

Therefore, R is symmetric.

For aRb , (a - b) is divisible by n.

For bRc, (b - c) is divisible by n.

Hence, (a - b) + (b-c) is divisible by n.

Or, (a-c) is divisible by n. This can be expressed as aRc.

Therefore, R is transitive.

So, R is an equivalence relation.

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infoexpert22

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