# Trying to reconcile function composition problems with sets & formulas

s3a
Homework Statement:
**Problem involving function composition with sets:**
Consider the set A = {w , x , y, z }, and the relations

S = {(w , x ), (w , y), (x , w ), (x , x ), (z , x )}

T = {(w , w ), (w , y), (x , w ), (x , x ), (x , z ), (y, w ), (y, y), (y, z )}

Find S ◦ T.

**Problem involving function composition with formulas:**
Let f(x) = x + 1 and g(x) = x^2.

Find f ◦ g.
Relevant Equations:
The answer for the set one is.:
S ◦T = {(w , w ), (w , x ), (w , z ), (w , y), (x , w ), (x , y), (x , x ), (x , z ), (z , w ), (z , x ), (z , z )}

The answer for the formula one is.:
f( g(x) ) = ( x^2 ) + 1

f( g(x) ) = x^2 + 1
I know how to solve each of those problems. For the set one, I look at the output of the S and try to match it with the input of T and then take the pair (input_of_S, output_of_T), and I do that for each pair.

As for the formula one, I just plug in x = g(y).

My confusion lies in trying to reconcile the two methods as different algorithms that are doing the same thing.

What bothers me is that for the formulaic one x_f = g(x_g), but for the set one, it seems to be to be output_of_S = input_of_T, which to translate that to the formulaic way, would be like saying f(x_f) = x_g instead.

Given that I get similar results, I'm assuming that I'm making a small mistake somewhere, but I'm not sure what it is.

Any input would be greatly appreciated!

P.S.
Sorry for the weird question. :P

Homework Helper
Gold Member
2022 Award
Homework Statement:: **Problem involving function composition with sets:**
Consider the set A = {w , x , y, z }, and the relations

S = {(w , x ), (w , y), (x , w ), (x , x ), (z , x )}

T = {(w , w ), (w , y), (x , w ), (x , x ), (x , z ), (y, w ), (y, y), (y, z )}

Find S ◦ T.

**Problem involving function composition with formulas:**
Let f(x) = x + 1 and g(x) = x^2.

Find f ◦ g.
Relevant Equations:: The answer for the set one is.:
S ◦T = {(w , w ), (w , x ), (w , z ), (w , y), (x , w ), (x , y), (x , x ), (x , z ), (z , w ), (z , x ), (z , z )}
That looks like ##T \circ S## to me.

• SammyS
Homework Helper
2022 Award
I know how to solve each of those problems. For the set one, I look at the output of the S and try to match it with the input of T and then take the pair (input_of_S, output_of_T), and I do that for each pair.

As for the formula one, I just plug in x = g(y).

My confusion lies in trying to reconcile the two methods as different algorithms that are doing the same thing.

$f \circ g$ means "do $g$, then do $f$ to the result". So in set notation with $g: X \to Y$ and $f: Y \to Z$ it would be $$\begin{split} f &= \{ (x, f(x)) : x \in Y \} \subset Y \times Z \\ g &= \{ (x, g(x)) : x \in X \} \subset X \times Y \\ f \circ g &= \{ (x, f(g(x))) : x \in X \} \subset X \times Z \end{split}$$

• SammyS