Let R be relation on I defined as mRn iff m- n- Check if R is an equivalence relation-

Given, relation R defined on set of integer as #160; mRn ⇔ m ≤ n Reflexive:R is reflexive because of the = sign in the inequality. #160;Symmetric:Let ...

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Standard XII

Mathematics

Transitive Relations

Let R be re...

Question

Let $R$ be relation on I defined as mRn iff $m \leq n .$ Check if $R$ is an equivalence relation.

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Solution

Given, relation R defined on set of integer as
$m R n \Leftrightarrow m \leq n$

Reflexive:
R is reflexive because of the = sign in the inequality.

Symmetric:
Let $m R n$
$\Rightarrow m \leq n$
It does not implies $n \leq m$
Hence, R is not symmetric.

Transitive:
Let $m, n, p \in I$
Let $m R n$ , $n R p$
$\Rightarrow m \leq n$ and $n \leq p$
$\Rightarrow m \leq p$
Hence, R is transitive.

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Let R be relation on I defined as mRn iff m≤n. Check if R is an equivalence relation.

Given, relation R defined on set of integer as mRn⇔m≤nReflexive:R is reflexive because of the = sign in the inequality. Symmetric:Let mRn⇒m≤nIt does not implies n≤mHence, R is not symmetric.Transitive:Let m,n,p∈I Let mRn, nRp⇒m≤n and n≤p⇒m≤pHence, R is transitive.

Let $R$ be relation on I defined as mRn iff $m \leq n .$ Check if $R$ is an equivalence relation.