Question

To determine Inverse relation for the given relation.

Short answer

The inverse relation for the given relation is \({R^{ - 1}} = {\mathop{\rm graph}\nolimits} \) of \({f^{ - 1}}\).

Step-by-step solution

Given 

The given relation \(R = \{ (a,f(a))\mid a \in A\} \)

The Concept of inverse relation

If (x, y)∈R, then (y, x)∈R-1 and vice versa. i.e., If R is from A to B, then R-1 is from B to A. Thus, if R is a subset of A x B, then R-1 is a subset of B x A.

Determine the Inverse relation

The relation\(R = \{ (a,f(a))\mid a \in A\} \)

Let\(R\)be a relation from set\(A\)to set\(B\).

Then,\({R^{ - 1}} = \{ (b,a)/(a,b) \in R\} \)

Suppose that the function\(f\)from\(A\)to\(B\)is a one-to-one correspondence and

\(\begin{array}{l}R = {\rm{ graph of }}f\\R = \{ (a,b)/b = f(a)\} \\R = \{ a,f(a)Va \in A\} \end{array}\)

Then,

\(\begin{array}{l}{R^{ - 1}} = \{ (b,a)/(a,b) \in R\} \\{R^{ - 1}} = \{ (b,a)/b = f(a)\} \\{R^{ - 1}} = \left\{ {(b,a)/{f^{ - 1}}(b) = a} \right\}\\{R^{ - 1}} = \left\{ {b,{f^{ - 1}}(b)/b \in B} \right\}\\{R^{ - 1}} = {\rm{ graph of }}{f^{ - 1}}\end{array}\)

Conclusion:

Therefore, the inverse relation for the given relation is \({R^{ - 1}} = {\mathop{\rm graph}\nolimits} \) of \({f^{ - 1}}\)