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-

Given that, ∀ a,b ∈ Z, aRb if and only if a b is divisible by n. Now, I. Reflexive aRa ⇒ (a a) is divisible by n, which is true for any integer a as '0' is di ...

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Standard XII

Mathematics

Equivalence Relation

Let n be a fi...

Question

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.

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Solution

Given that, $\forall a, b \in Z$ , aRb if and only if a - b is divisible by n.
Now,
I. Reflexive
$a R a \Rightarrow (a - a)$ is divisible by n, which is true for any integer a as '0' is divisible by any positive integer.
Hence, R is reflexive.

II. Symmetric
aRb
$\Rightarrow a - b$ is divisible by n.
$\Rightarrow - (b - a)$ is divisible by n.
$\Rightarrow (b - a)$ is divisible by n.
$\Rightarrow b R a$
Hence, R is symmetric.

III. Transitive

Let $a R b a n d b R c$
$\Rightarrow (a - b)$ is divisible by n and $(b - c)$ is divisible by n
$\Rightarrow (a - b) + (b - c)$ is divisibly by n
$\Rightarrow (a - c)$ is divisible by n
$\Rightarrow a R c$
Hence, R is transitive.

So, R is an equivalence relation.

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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.