# Thread: Functions and Inverses

1. ## Functions and Inverses

I am not exactly sure what these functions mean:

$\overrightarrow{f}: \mathcal{P}(X) \rightarrow \mathcal{P}(Y)$ defined by $\overrightarrow{f}(A) = \{ f(x) | x \in A \}$ for $A \in \mathcal{P}(X)$ and

$\overleftarrow{f}: \mathcal{P}(Y) \rightarrow \mathcal{P}(X)$ defined by $\overleftarrow{f}(B) = \{x \in X | f(x) \in B \}$ for $B \in \mathcal{P}(Y)$.

These two functions really are $f$ and $f^{-1}$ right?

So if $f$ is a bijection with inverse $f^{-1}$, then $f(x) = y_0$ if and only if $x = f^{-1}(y_0)$ so that $\overleftarrow{f}(\{y_0 \}) = \{f^{-1}(y_0) \}$. So this implies that $\overleftarrow{f}(\{y_0 \}) \neq \{f^{-1}(y_0) \}$ sometimes?

2. Originally Posted by tukeywilliams
$\overrightarrow{f}: \mathcal{P}(X) \rightarrow \mathcal{P}(Y)$ defined by $\overrightarrow{f}(A) = \{ f(x) | x \in A \}$ for $A \in \mathcal{P}(X)$ and

$\overleftarrow{f}: \mathcal{P}(Y) \rightarrow \mathcal{P}(X)$ defined by $\overleftarrow{f}(B) = \{x \in X | f(x) \in B \}$.

These two functions really are $f$ and $f^{-1}$ right?
No, they are really two new functions related to the function $f$.
The two new functions are mapping of subsets to subsets.
Whereas $f$ maps set X to set Y.

3. Yes, but why do we need these functions? Are they only useful for figuring out whether functions have inverses?

In other words, if $\overleftarrow{f}(\{y_0\}) \neq \{f^{-1}(y_0)\}$ then $f$ has no inverse?

4. Originally Posted by tukeywilliams
Yes, but why do we need these functions?
This inverse image function is really useful.

I do not think you taken group theory, given the homomorphism $\phi: G \mapsto G'$ then $\phi ^{-1} [\{e\}]$ where $\{e'\}\subseteq G'$ is called the kernel it is extremely important concept.

Have you even taken linear algebra? Given a linear transformation $T: V\to V'$ then $\ker T$ is defined exactly like above $T^{-1} [ \bold{0}]$ where $\bold{0}$ is the zero vector of $V'$. This is another important concept.

5. Yes I have taken linear algebra.

6. Originally Posted by tukeywilliams
Yes I have taken linear algebra.
Do you understand what I mean by the kernel of a linear transformation. Carefully try to understand that the kernel is defined as an inverse image.

7. Yeah the kernel of a linear transformation is the set of vectors v such that $T(v) =0$, i.e. the set of vectors that maps to the 0 vector.

So the inverse image function takes the kernel and maps it to the singleton set $\{0 \}$ and vice versa.

8. Originally Posted by tukeywilliams
Yes, but why do we need these functions?

In other words, if $\overleftarrow{f}(\{y_0\}) \neq \{f^{-1}(y_0)\}$ then $f$ has no inverse?
The answer to the first question is: it is a foundational issue. That is, in a course on the foundation of mathematics, logic & sets, a function is defined as a relation with additional properties. If $f$ is a function from A to B then $f$ is also a relation from A to B. Now it is true then that $f^{-1}$ is also a relation from A to B BUT it may not be a function.

However $\overleftarrow f$ is always a function from $P(B)$ to $P(A)$.