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.

Could someone please help me reconcile the two approaches (to similar problems)?

Any input would be greatly appreciated!

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

 

[PeroK]

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.

 

[pasmith]

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 $$\begin{split} f &= \{ (x, f(x)) : x \in Y \} \subset Y \times Z \\ g &= \{ (x, g(x)) : x \in X \} \subset X \times Y \\ f \circ g &= \{ (x, f(g(x))) : x \in X \} \subset X \times Z \end{split}$$

 

[WWGD]

Consider that , for one , , while every function is a relation, the converse doesn't hold.

You can consider a set R in your case as a relation r, where (a,b) stands for (a,r(a)), i.e., b:=r(a).