Question: Let \(X=\left\{ 1,2,3,4,5 \right\}\) and \(A=\left\{ a,b,c,d \right\}\). Let \(f:X\to A\) be the function defined by \(f\left( 1 \right)=f\left( 2 \right)=d\), \(f\left( 3 \right)=b\), \(f\left( 4 \right)=f\left( 5 \right)=a\).

(a) List the elements of the Cartesian product \(X\times A\).

(b) Highlight the elements of \(X\times A\) which are in the graph \({{\Gamma }_{f}}\) of f.

(c) Is the function f injective? Why?

(d) Is the function f surjective? Why?

(e) List the elements of each of the fibres of f over the elements of A.

(f) Let \(g:X\to A\) be the function with the same values as f except that g(5) = c. Explain why g is surjective and find two different right inverses for g.