Characteristic Function Comprehesnion

Question

I am having trouble wrapping my head around characteristic functions. I'm not the best when it comes to taking theoretical explanations and then applying it to an actual problem. I'm a hand's-on learner and really need to see something and repeat the steps myself to really grasp wtf I'm even doing. Really what I need is a super dumbed down walk-through. So if anyone can help me with the following, I would be appreciative. (Keep in mind this IS a homework assignment and I would much rather learn what's going on than just be shown an answer because that doesn't help me.)

---

Define $f: \Bbb R → \{0,1\}$ as the characteristic function of $\Bbb Z+$, the set of positive integers. That is, $f(x) = \chi_{A}(x)$, for the set $A$ of positive integers $= \{ n | n \in \Bbb Z+ \}$.

Answer the following questions using the definition of $f$ given above. Your answers for (b) and (c) should each be a set. You may specify each set by listing the values, using set builder notation, or describing the set in words.

a) Draw the graph of the function $f(x)$ for $-2 \leq x \leq 2$.

b) What is $f(\{x \mid −1 < x < 4\}$? That is, what set is the image of the set of values between $-1$ and $4$? Explain.

c) What is $f^{-1}( \{x \mid 2 < x < 5 \}$? That is, what set is the preimage of the set of values between $2$ and $5$? Explain.

Answer 1

So if $A$ is any set, then $\chi_{A}(x)$ is the function defined by $$\chi_{A}(x) = \begin{cases} 1 & x \in A \\ 0 & x \not \in A \end{cases}. $$

For example, take the set $A = \{1,5\}$ of two elements. Then if we let the domain of the function $\chi_{\{1,5\}}(x)$ be the reals $\Bbb R$, we have:

$\chi_{\{1,5\}}(1) = 1$, $\chi_{\{1,5\}}(\pi) = 0$, $\chi_{\{1,5\}}(2.16) = 0$, $\chi_{\{1,5\}}(5) = 1$, $\chi_{\{1,5\}}(10000) = 0$.

$\chi_{\{1,5\}}(x)$ equals $1$ only if $x$ is in the set $\{1,5\}$, i.e., only if the input $x$ is either $1$ or $5$. For any other input other than $1$ or $5$, $\chi_{\{1,5\}}(x)$ outputs $0$.

In your case, $A = \{1,2,3,4,\dots\}$.

So $\chi_{\{1,2,3,4,\dots\}}(x)$ is the function that equals $1$ when the input is in the set $\{1,2,3,4,\dots\}$ and equals $0$ otherwise. From this description, I think it should be easy for you to draw this function's graph.

Answer 2

The characteristic function for a set $A \subseteq \mathbb R$ is a function so that $\chi_A(x)=\begin{cases}1 & \textrm{if}\,\, x \in A\\ 0 & \textrm{otherwise} \end{cases}$

It's pretty much just "indicating" when $x$ is in a set.

$\chi_{\mathbb{Z}}(x)$ is $1$ if $x$ is an integer, and zero otherwise.

So, for the first problem: $\chi_{\mathbb{Z}}:\mathbb{R} \to \mathbb{R}$ is zero if $x \notin \mathbb{Z}_+$. What are $\chi_{\mathbb{Z}}(0)$ or $\chi_{\mathbb{Z}}(.5)$, for example? What is the image of $\chi_{\mathbb Z}([-2,2])$? Well, definitely $0$ for all $x \neq 1,2$, and by definition $1$ for $x=1,2$.