# Union and Intersection proof

• Sep 30th 2012, 05:41 PM
fallenstars16
Union and Intersection proof
For a set of Y real numbers define a function CY by;

CY(x)= {1 if x ∈Y,
0 if x is not ∈ Y.

Let Y and Z be a set of real numbers, find expressions for CYUZ and Cynz
YUZ={x∈R|x∈Y or x∈Z}
ynz={x∈R|x∈Y and x∈Z}

so i know what Cy is but I'm unsure of what Cz is and what I am suppose to do. Help would be greatly appreciated.
• Sep 30th 2012, 09:25 PM
FernandoRevilla
Re: Union and Intersection proof
Using the given definitions, you'll obtain

$\displaystyle (i)\;\;\;C_{Y\cap Z}=C_Y\cdot C_Z$
$\displaystyle (ii) \;\;C_{Y\cup Z}=C_Y+C_Z-C_Y\cdot C_Z$

For example:

$\displaystyle x\in Y\cap Z\Rightarrow x\in Y\mbox{ and } x \in Z \Rightarrow \begin{Bmatrix} C_{Y\cap Z}(x)=1\\C_{Y}(x)=1\\C_{Z}(x)=1\end{matrix}\\ \Rightarrow C_{Y\cap Z}(x)=C_Y(x)\cdot C_Z(x)$

$\displaystyle x\not\in Y\cap Z\Rightarrow \ldots$

(try the rest).
• Oct 1st 2012, 04:03 AM
emakarov
Re: Union and Intersection proof
Quote:

Originally Posted by FernandoRevilla
Using the given definitions, you'll obtain

$\displaystyle (i)\;\;\;C_{Y\cap Z}=C_Y\cdot C_Z$
$\displaystyle (ii) \;\;C_{Y\cup Z}=C_Y+C_Z-C_Y\cdot C_Z$

Alternatively,

$\displaystyle C_{Y\cap Z}(x)=\min(C_Y(x), C_Z(x))$
$\displaystyle C_{Y\cup Z}(x)=\max(C_Y(x),C_Z(x))$.