# Characteristic Function on A

• Nov 15th 2011, 06:26 PM
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.
• Nov 15th 2011, 06:48 PM
Drexel28
Re: Characteristic Function on A
Quote:

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 $\displaystyle A,IB$?
• Nov 15th 2011, 07:02 PM
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
• Nov 16th 2011, 04:03 AM
emakarov
Re: Characteristic Function on A
Quote:

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 $\displaystyle \chi_A(\chi_B(x))$?

Quote:

b.)Sketch the graph f|A•IB

Quote:

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.
• Nov 16th 2011, 12:01 PM
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. (Crying)
• Nov 16th 2011, 01:24 PM
emakarov
Re: Characteristic Function on A
Quote:

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.
• Nov 16th 2011, 03:39 PM
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. ??
• Nov 16th 2011, 03:48 PM
emakarov
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 $\displaystyle x$?
• Nov 16th 2011, 03:54 PM