Union and Intersection proof

**fallenstars16** · September 30th 2012, 05:41 PM

For a set of Y real numbers define a function C_Y by;

C_Y(x)= {1 if x ∈Y,
0 if x is not ∈ Y.

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

so i know what C_y is but I'm unsure of what C_z is and what I am suppose to do. Help would be greatly appreciated.

**FernandoRevilla** · September 30th 2012, 09:25 PM

Using the given definitions, you'll obtain

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

For example:

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

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

(try the rest).