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

    \[\forall \mathbf{a},\text{ }\mathbf{b}\in \mathbf{Z}\]

, aRb if and only if a – b is divisible by n. Show that R is an equivalance relation.
CBSE | Class 12 | Exercise 1 | Maths | NCERT Exemplar | Relations and Functions
Let n be a fixed positive integer. Define a relation R in Z as follows:

    \[\forall \mathbf{a},\text{ }\mathbf{b}\in \mathbf{Z}\]

, aRb if and only if a – b is divisible by n. Show that R is an equivalance relation.

Given

    \[\forall \mathbf{a},\text{ }\mathbf{b}\in \mathbf{Z}\]

, aRb if and only if a – b is divisible by n.

Now, for

    \[aRa\Rightarrow \left( a\text{ }\text{ }a \right)\]

is divisible by n, which is true for any integer a as ‘

    \[0\]

’ is divisible by n.

Thus, R is reflective.

Now, aRb

So, (a – b) is divisible by n.

⇒ – (b – a) is divisible by n.

⇒ (b – a) is divisible by n

⇒ bRa

Thus, R is symmetric.

Let aRb and bRc

Then, (a – b) is divisible by n and (b – c) is divisible by n.

So, (a – b) + (b – c) is divisible by n.

⇒ (a – c) is divisible by n.

⇒ aRc

Thus, R is transitive.

So, R is an equivalence relation.

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