# Thread: Characteristic Function on A

1. ## Characteristic Function on A

Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=(2,5].

a.)Sketch the graph of XA•XB

(which has X with an A subscript and X with B subscript, i suppose means XA composite XB)

b.)Sketch the graph f|A•IB

Now i know how to graph XA and XB but im not sure what to do with a.) and i dont know what b.) is even asking.

Any help or suggestions. Much appreciated.

2. ## Re: Characteristic Function on A

Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=(2,5].

a.)Sketch the graph of XA•XB

(which has X with an A subscript and X with B subscript, i suppose means XA composite XB)

b.)Sketch the graph f|A•IB

Now i know how to graph XA and XB but im not sure what to do with a.) and i dont know what b.) is even asking.

Any help or suggestions. Much appreciated.
Try to prove (or note) that in general that, in your notation, $\displaystyle \chi_A\chi_B=\chi_{A\cap B}$. I don't know what they're asking for b) either, what does the vertical line indicate? What are $\displaystyle A,IB$?

3. ## Re: Characteristic Function on A

XAºXB i suppose i should have written it as. XA composite XB. I'm not sure if its the intersection. IB i think is the Identity function: B-->B

4. ## Re: Characteristic Function on A

Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=(2,5].

a.)Sketch the graph of XA•XB

(which has X with an A subscript and X with B subscript, i suppose means XA composite XB)
For any x, $\displaystyle (\chi_A\circ\chi_B)(x)=\chi_A(\chi_B(x))$. Now, $\displaystyle \chi_B(x)$ returns either 0 or 1, so what's the value of $\displaystyle \chi_A(\chi_B(x))$?

b.)Sketch the graph f|A•IB
IB i think is the Identity function: B-->B
In this case, it looks like $\displaystyle f|_A\circ I_B$, i.e., the composition of the restriction of $\displaystyle f$ on $\displaystyle A$ and the identity function on $\displaystyle B$. So, $\displaystyle (f|_A\circ I_B)(x)=f|_A(I_B(x))=f|_A(x)=2-3x$. Since $\displaystyle I_B$ is defined on (2, 5], the question now is only to determine for which $\displaystyle x\in(2,5]$ the function application $\displaystyle f|_A(x)$ is defined.

5. ## Re: Characteristic Function on A

Well for part a. would (2,5] be on 1? So would it just be as if we were graphing just XB? Cause if i put XB inside XA which is what the composite is wouldnt it just be the intersection of XA and XB?

Its not making sense to my pea brain.

6. ## Re: Characteristic Function on A

Originally Posted by emakarov
For any x, $\displaystyle (\chi_A\circ\chi_B)(x)=\chi_A(\chi_B(x))$. Now, $\displaystyle \chi_B(x)$ returns either 0 or 1, so what's the value of $\displaystyle \chi_A(\chi_B(x))$?
If I understand the definitions correctly,

$\displaystyle \chi_A(x)=\begin{cases}1&x\in[-1,3)\\0&\mbox{otherwise}\end{cases}$

$\displaystyle \chi_B(x)=\begin{cases}1&x\in(2,5]\\0&\mbox{otherwise}\end{cases}$

Case 1: $\displaystyle x\in(2,5]$. Then $\displaystyle \chi_B(x)=1$, so $\displaystyle \chi_A(\chi_B(x))=\dots$

Case 2: $\displaystyle x\notin(2,5]$. Then $\displaystyle \chi_B(x)=0$, so $\displaystyle \chi_A(\chi_B(x))=\dots$

As an aside note, one of the confusing things about $\displaystyle \chi_A(\chi_B(x))$ is the different meaning given to real numbers. The output of $\displaystyle \chi_B(x)$ is 0 or 1, yet it is interpreted as any other real number and is fed to $\displaystyle \chi_A$. In programming, it would be clearer to make $\displaystyle \chi_A$ and $\displaystyle \chi_B$ return Boolean values True and False instead of numbers 1 and 0. Then $\displaystyle \chi_A(\chi_B(x))$ would be ill-typed.

7. ## Re: Characteristic Function on A

The syntax and math jargon of this question is probably whats making this difficult to me.

So what does it mean for instance in case1: XA(1) and case 2: XA(0).?

For case 1: Would i graph the line then at y=1, x=[ -1,3) for XA=1. ??

8. ## Re: Characteristic Function on A

Before sketching anything, you need to learn to compute compositions of functions. So, what are $\displaystyle \chi_A(1)$ and $\displaystyle \chi_A(0)$, and therefore, what is $\displaystyle \chi_A(\chi_B(x))$ for all $\displaystyle x$?

9. ## Re: Characteristic Function on A

i understand the composition of functions and relations , but A and B are just sets arent they? This isnt making any sense. The characteristic functions aren't given much explanation in my textbook. So i have no idea what there purpose even is. It's just a review problem our teacher gave us and I'm just trying to graph it.