Types of relation of a set, Equivalence Relation & Congruence Modulo Relation of Integers

Mathematics Class XI Chapter 2: Relations & Functions Types of relation of a set, Equivalence Relation & Congruence Modulo Relation of Integers

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Types of relation of a set

A relation $R$ on a set $A$ is called

(i) Reflexive:

$aRa , \forall a\in A (\text {i.e.,} (a,a) \in R , \text { for every } a \in A)$

(ii) Symmetric:

If $aRb \implies bRa, a ,b \in A$

$( \text {i.e.,} (a,b) \in R \implies (b,a) \in R, \text { for } a, b \in A)$

(iii) Transitive:

If $aRb$ and $bRc \implies aRc, a,b,c \in A$

$( \text {i.e.,} (a,b)\in R$ and $(b,c) \in R \implies (a,c) \in R$ ; for $a,b,c \in A)$

Equivalence Relation:

A relation $R \subseteq A \times A$ is called an equivalence relation on $A$ if it is

(i) reflexive, (ii) symmetric, (iii) transitive

It is symbolically represents ‘ $\sim$ ‘.

Congruence Modulo Relation on Integers:

Let $a, b\in Z$ and $m$ be a fixed integer. We say that ‘ $a$ is congruent to $b$ modulo $m$ ‘ and written as $a\equiv b ( \text { mod } m ) \text { iff}$ $m$ divides $a-b$ .

Note:

In this case $a$ is of the form $a =b+mk$ for some $K \in Z$ .