Composition of functions with Venn diagrams

Text

Two functions are given via Venn diagrams.

• The function $g$ sends the set $\{1,2,3,4,5\}$ to the set of icons $\{\text{sun},\text{lightning},\text{tree},\text{coffee},\text{heart}\}$ with: $1\to\text{sun}$, $2\to\text{tree}$, $3\to\text{tree}$, $4\to\text{coffee}$, $5\to\text{heart}$.

• The function $f$ sends the set $\{A,B,C,D\}$ to $\{1,2,3,4,5\}$ with: $A\to 2$, $B\to 3$, $C\to 2$, $D\to 5$.

• (a) Represent the function $h=g\circ f$ in a further diagram.

• (b) Determine the image set of $f$ and that of $g$.

(diagrams in the original test)

Solution

(a) By definition $h(x)=g\big(f(x)\big)$ (apply $f$ first, then $g$): $h(A)=g(2)=\text{tree},\quad h(B)=g(3)=\text{tree},\quad h(C)=g(2)=\text{tree},\quad h(D)=g(5)=\text{heart}.$

(b) The image set collects only the values actually reached: $\operatorname{Im}(f)=\{2,3,5\},\qquad \operatorname{Im}(g)=\{\text{sun},\text{tree},\text{coffee},\text{heart}\}.$ (Nobody reaches $\text{lightning}$, so $g$ is not surjective; moreover $2$ and $3$ share the same image under $g$, so $g$ is not injective.) Consistently $\operatorname{Im}(h)=\{\text{tree},\text{heart}\}$.