How would I go about answering this question about functions?

Question

Hey so I have this exercise and I don't know how to tackle it or present my answer;

For $A : = \{1, 2, 3, 4\}$, $B := \{a, b, c, d\}$ and $C := \{1, 2, 3\}$, let $f : A \to B$ be the function $\{(1, a),(2, c),(3, b),(4, d)\}$, and let $g : B \to C$ be the function $\{(a, 1),(b, 2),(c, 3),(d, 1)\}$.

Determine the function $g \circ f : A \to C$.

Can anyone give me some guidance? Please. I haven't really started it yet, because I don't know where.

Would it just be y $(g \circ f)(x) = g(f(x))$ or do I have to write out the sets?

Answer 1

Given that the task is quite basic, I believe you should write out the sets. That is, you should write out the function $g\circ f$ as $$g\circ f = \{(1, *), (2, *), (3, *), (4, *)\}$$

with appropriate values in place of the asterisks. To calculate what value should be put next to $1$, for example, you can use the rule

$$(g\circ f)(1) = g(f(1))$$

and first calculate $f(1)$, then calculate $g(*)$, where you put whatever $f(1)$ is in place of the asterisk.

Answer 2

Take some element of $A$, for instance $1$.

Observe that $f$ sends this element to $a\in B$ because $(1,a)\in f$.

Now observe that $g$ on its turn sends $a$ to element $1\in C$ because $(a,1)\in g$.

This means that $g\circ f$ will send the original element $1\in A$ to element $1\in C$.

This can be rephrased by saying that $(1,1)\in g\circ f$.

Now do the same for the other elements of $A$ and you will end up with something like $$g\circ f=\{(1,1),(2,\cdot),(3,\cdot),(4,\cdot)\}$$