Question:
Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.
Solution:
The functions $f:\{1,3,4\} \rightarrow\{1,2,5\}$ and $g:\{1,2,5\} \rightarrow\{1,3\}$ are defined as
$f=\{(1,2),(3,5),(4,1)\}$ and $g=\{(1,3),(2,3),(5,1)\}$
$\begin{array}{ll}g \circ f(1)=g(f(1))=g(2)=3 & {[f(1)=2 \text { and } g(2)=3]} \\ g \circ f(3)=g(f(3))=g(5)=1 & {[f(3)=5 \text { and } g(5)=1]} \\ g \circ f(4)=g(f(4))=g(1)=3 & {[f(4)=1 \text { and } g(1)=3]} \\ \therefore g \circ f=\{(1,3),(3,1),(4,3)\} & \end{array}$