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

Inverse Relation

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

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