Question

Let X = {a, b, c} defined by f : X ®X such that f = {(a, b), (b, a), (c,c)}.  Find the values of

f^–1, f² and f⁴. 

Answer 1

Let $X = \{a, b, c\}$ defined by $f : X \to X$ such that $f = \{(a, b), (b, a), (c,c)\}$ .

Let us find the values of $f^{–1}, f^2$ and $f^4$. Since $f^{-1}(y)=x$ iff $f(x)=y,$ we concluse that $f^{-1} = \{(b, a), (a, b), (c,c)\}.$ Taking itno aaccount that $f^2(x)=f(f(x))$ and $f^4(x)=f^2(f^2(x)),$ we conclude that $f^2 = \{(a, a), (b, b), (c,c)\}$ and $f^4 = \{(a, a), (b, b), (c,c)\}.$