Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

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

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 .

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