Relation and Function

tags: Mathematics

All Mathematics Formula by Abhyas here

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Relation

Let

$A$ and

$B$ be two sets then Relations

$R$ from

$A$ to

$B$ is a subset of

$A\times B$ i.e.

$R\subseteq A\times B$

Also, If

$a\in A$ and

$b\in B$ then,

$R=\{(a,b):a\in A$ and

$b\in b\}$

This can be denoted by

$xRy\iff$ condition where

$x\in A$ and

$y\in B$

e.g. Let

$A=a,b,c,d$ and

$B=1,2,3,4$

Let,

• $R_1=\{(b,3),(b,4),(a,3)\}$ is a relation
• $R_2=\{(a,1)\}$ is a relation
• $R_3=\{(a,1),(b,2),(c,3),(d,4)\}$ is a relation
• $a{R}_{4}b\iff x$ divides
  $y$
• $R_5=\{(a,4),(c,1),(3,d)\}$ is not a relation,
  $\because (3,d)\notin A\times B$
• $R_6=\{(1,a),(2,c),(3,d)\}$ is not a relation,
  $\because R_5⊈A\times B$

Domain, co-domain and Range of

$R$

Domain

Set of all the first element of order pair in relation

$R$.

$\therefore$ Domain of

$R=\{a:(a,b)\in R\}$

Range

Set of all the second element of order pair in relation

$R$.

$\therefore$ Domain of

$R=\{b:(a,b)\in R\}$

Codomain

Set of all elements of set

$B$.

$a\in A$

Total number of Possible Relations

Let

$n(A)=m$ and

$n(B)=n$

$\because n(A\times B)=mn$

$\therefore$ Total number of possible relations

$=2^{mn}$ (This is same as number of subsets of

$A\times B$)

Licensing and Links

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