Vtyjkyuk

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?

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

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

$$(gcirc f)(1) = g(f(1))$$

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

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

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

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

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

This can be rephrased by saying that $(1,1)in gcirc f$.

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