Inverse Relation

Mathematics Class XI Chapter 2: Relations & Functions Inverse Relation

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The inverse of $f$ , written as $f^{-1}$ , is a relation from $B$ to $A$ defined by $f^{-1}= \{(y,x): (x,y) \in f \}$ .

$\rightarrow$ It is obvious that $\text {dom f} (D_f)$ is a subset of $A$ and $\text { rng f } (R_f)$ is a subset of $B$ .

$\rightarrow$ It is also clear that $f^{-1}$ is a subset of $B$ and range of $f^{-1}$ is a subset of $A$ .

Example:

Let $A =\{ \text{Dasaratha, Rama, Laxman, Sita, Janaka} \}$

Taking $B=A$ , then

(i) The relation Fatherhood on $A$ is

$R_1=\{ (\text{Dasaratha, Rama)}, (\text {Dasaratha, Laxman)(Janaka, Sita) }\}$

(ii) The daughterhood relation on $A$ is

$R_2=\{ \text{(Sita, Janaka ) } \}$

(iii) The sonhood relation on $A$ is

$R_3=\{ \text{(Rama, Dasaratha )}, \text{(Laxman, Dasaratha )}\}$

(iv) The brotherhood relation on $A$ is

$R_4=\{ \text{(Rama, Laxman )}, \text{(Laxman, Rama )}\}$

$\begin{aligned} \rightarrow \text {dom } R_1&=\{ \text{(Dasaratha, Janaka)}, \\ \text {rng}R_1&= \{\text {Rama, Laxman, Sita}\}\end{aligned}$

$R_1^{-1}= \{( \text {Rama, Dasaratha}), ( \text {Laxman, Dasaratha}), ( \text {Sita, Janaka})\}$

$\rightarrow\text {dom}R_2= \{ \text {Sita}\}, \text {rng}R_2 = \{\text {Janaka}\}$

$R_2^{-1}=\{ \text {Janaka, Sita}\}$

$\begin{aligned}\rightarrow\text {dom } R_3&=\{\text { Rama, Laxman}\}\\ \text{rng}R_3&=\{ \text {Dasaratha}\}\\R_{3}^{-1}&= \{(\text{Dasaratha, Rama}), (\text {Dasaratha, Laxman})\}\end{aligned}$

$\begin{aligned} \rightarrow\text { dom }R_4&= \{\text {Rama, Laxman}\}\\ \text{rng } R_4&=\{\text {Laxman, Rama}\}=\text {dom }R_4\end{aligned}$

$R_4^{-1}=\{(\text {Laxman, Rama}), )(\text {Rama, Laxman})\} = R_4$