# Trying to reconcile function composition problems with sets & formulas

• s3a
In summary: The inverse of a is just the set of all elements that are not in a. So (a,b) in the above example would be (x, not-a), (y, not-b), and so on. In summary, the two approaches seem to be doing the same thing, but reconciling their differences is causing confusion for the author.
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

s3a said:
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
s3a said:
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
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).

## What is function composition in mathematics?

Function composition is the process of applying one function to the results of another function. If you have two functions, f(x) and g(x), the composition is written as (f ∘ g)(x) = f(g(x)). This means you first apply g to x, and then apply f to the result of g(x).

## How do you reconcile function composition with set theory?

In set theory, functions are viewed as sets of ordered pairs. To reconcile function composition with set theory, you can think of the composition (f ∘ g) as a set of ordered pairs where each element in the domain of g maps to an element in the domain of f through g, and then that result maps to an element in the codomain of f. Formally, (f ∘ g) = {(x, z) | ∃y (x, y) ∈ g and (y, z) ∈ f}.

## What are common pitfalls when solving function composition problems?

Common pitfalls include not correctly applying the order of operations, misunderstanding the domains and codomains of the functions involved, and incorrectly simplifying the composed function. It is crucial to follow the correct order: apply the inner function first and then the outer function, and ensure that the output of the inner function is within the domain of the outer function.

## How can you use formulas to solve function composition problems?

To use formulas in solving function composition problems, write down the explicit formulas for each function involved. Substitute the formula of the inner function into the formula of the outer function. Simplify the resulting expression to find the formula for the composed function. For example, if f(x) = 2x + 3 and g(x) = x^2, then (f ∘ g)(x) = f(g(x)) = f(x^2) = 2(x^2) + 3.

## Can function composition be visualized using diagrams?

Yes, function composition can be visualized using diagrams such as mapping diagrams or arrow diagrams. In these diagrams, elements from the domain of the first function (g) are mapped to elements in its codomain, which is also the domain of the second function (f). The elements in the codomain of g are then mapped to elements in the codomain of f, illustrating the overall mapping from the domain of g to the codomain of f.

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