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 \(f\left( 1 \right)=f\left( 2 \right)=d\), \(f\left( 3 \right)=b\), \(f\left( 4 \right)=f\left( 5 \right)=a\).

1. List the elements of the Cartesian product \(X\times A\).
2. Highlight the elements of \(X\times A\) which are in the graph \({{\Gamma }_{f}}\) of f.
3. Is the function f injective? Why?
4. Is the function f surjective? Why?
5. List the elements of each of the fibres of f over the elements of A.
6. 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.

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