Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$

Question

I haven't been able to do this exercise:

Let $f: A \rightarrow B$ be any function. $f^{-1}(X)$ is the inverse image of $X$. Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ where $X \subseteq B$.

Since $X \subseteq B$, all the elements in $X$ belong to the codomain of $f$.

Since $f$ is surjective, it means that all elements in the codomain $B$ have some preimage in $A$. Given that $X \subseteq B$, all elements in $X$ must also have a preimage in $A$.

Have $\triangle = f^{-1}(X)$, $\triangle$ is now a set containing the preimages of the elements in $X$. Because of this, $\triangle \subseteq A$.

If we evaluate $f(\triangle)$, we...... nope, I don't know what I'm doing now.

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What do you think?

Answer 1

In this cases double inclusion is the way: let $x \in f(f^{-1}(X))$, then there is $a \in f^{-1}(X)$ such that $f(a) = x$. By definition of the pre image, $x \in X$. This gives $$f(f^{-1}(X)) \subset X.$$ Note that we didn't use that the function is surjective to prove this inclusion, meaning that it holds in general.

Now let $x \in X$. Being $f$ surjective, we can find $a \in A$ such that $f(a) = x$. This gives $a \in f^{-1}(X)$ and hence $x = f(a) \in f(f^{-1}(X))$. This shows $$X \subset f(f^{-1}(X)).$$

Answer 2

Here is how I would do this: go to the element level, expand the definitions and basic properties, and use logic to simplify. Start with the most complex expression, which is here $\;f[f^{-1}[X]]\;$.

So for any $\;b\;$, \begin{align} & b \in f[f^{-1}[X]] \\ \equiv & \;\;\;\;\;\text{"basic property of $\;\cdot[\cdot]\;$"} \\ & \langle \exists a : a \in f^{-1}[X] \;\land\; f(a) = b \rangle \\ \equiv & \;\;\;\;\;\text{"basic property of $\;\cdot^{-1}[\cdot]\;$"} \\ & \langle \exists a :: f(a) \in X \;\land\; f(a) = b \rangle \\ \equiv & \;\;\;\;\;\text{"logic: use right conjunct in left to substitute"} \\ & \langle \exists a :: b \in X \;\land\; f(a) = b \rangle \\ \equiv & \;\;\;\;\;\text{"logic: extract conjunct which does not use $\;a\;$ out of $\;\exists a\;$"} \\ & b \in X \;\land\; \langle \exists a :: f(a) = b \rangle \\ \equiv & \;\;\;\;\;\text{"assumption: $\;f\;$ is surjective, i.e., $\;\langle \forall b :: \langle \exists a :: f(a) = b \rangle \rangle\;$"} \\ & b \in X \\ \end{align} By set extensionality this proves the statement in question.

Strongly related: the proof of (4) in an answer of mine to another question (https://math.stackexchange.com/a/434230/11994).

Answer 3

I thought to provide some intuition by virtue of a picture. As drawn, $f$ is NOT surjective, because:

$a \in$ set X. $b, c \in$ set $Y.$ Your $X$ is my $S$ and $S \subsetneq Y$ here.

The pink bespeaks both $f^{-1}(S)$ and $f: X \to Y$ defined by $f(a) = b$. The green point is $c \in Y$.

Then $f(f^{-1}(\{c\}))=f(\varnothing)=\varnothing\neq\{c\}\quad\text{and}\quad f(f^{-1}(Y))=f(\{a\})=\{b\}\neq Y$.

Yet, espy that if I extend/stretch $f(X)$ (in pink inside $S$) to "cover" all of $Y$, then $f(X) = Y$.
By virtue of the definition of surjectivity, this mans $f$ is surjective.