Find $f\circ g$, where $f, g : A \to A$ with $A=\{x, y, z\}$

Question

I have been solving similar questions and got them right, except this one where the answer in the back of the book doesn't match mine. The question goes:

Exercise. Find $f\circ g$, where $f, g : A \to A$ with $A=\{x, y, z\}$ and $f,g$ are defined by $f = \{(x,y),(y,z),(z,x)\}$ and $g = \{(x,y),(y,x),(z,z)\}$.

Answer in the back says, it's $\{(x,x),(y,z),(z,y)\}$ whereas I'm getting $\{(x,z),(y,y),(z,x)\}$. Could someone please help?

Answer 1

If $f\circ g=f(g(x))$, then

• $x\xrightarrow{g}y\xrightarrow{f}z$,
• $y\xrightarrow{g}x\xrightarrow{f}y$,
• $z\xrightarrow{g}y\xrightarrow{f}x$,

and hence $f\circ g=\{(x,z),(y,y),(z,x)\}$. And if $f\circ g=g(f(x))$, then

• $x\xrightarrow{f}y\xrightarrow{g}x$,
• $y\xrightarrow{f}z\xrightarrow{g}z$,
• $z\xrightarrow{f}y\xrightarrow{g}y$,

and hence $f\circ g=\{(x,x),(y,z),(z,y)\}$. So the result depends on the definition of $f\circ g$. By the way, we usually define $f\circ g=f(g(x))$.