Hi,

first of all, Merry Christmas! I am doing some math while digesting another huge dinner, and have a small question.

Problem:

If A is any set of real numbers, define a function $\displaystyle C_A$ as follows:

$\displaystyle C_{A}(x)=\left\{ \begin{array}{cc}

1, & \mbox{x in A}\\

0, & \mbox{x not in A}\\

\end{array} \right.

$

Find expressions for $\displaystyle C_{A\cap B}$ and $\displaystyle C_{A\cup B}$ and $\displaystyle C_{\Re-A}$

Attempt:

The way I am reading this, is that I first have to define a function which is 1 for real numbers, and 0 for pure imaginary numbers.

I think this one works fine:

$\displaystyle C_{A}(x)=\frac{x+\overline{x}}{2x} $ where $\displaystyle \overline{x}$ is the complex conjugate.

I am not so sure about the other expressions. If I still take B to be the set of pure imaginary numbers am I suppose to define the following functions:

$\displaystyle C_{A\cap B}(x) = \left\{ \begin{array}{cc}

0, & x\in A \\

0, & x\in B \\

1, & x\in B \quad and \quad x\in A\\

\end{array}\right. $

$\displaystyle C_{A\cup B}(x) = \left\{ \begin{array}{cc}

1, & x\in A \quad and \quad x\in B\\

1, & x\in A\\

1, & x\in B\\

\end{array}\right. $

$\displaystyle C_{\Re-A}(x) = \left\{ \begin{array}{cc}

0, & x\in A\\

0, & x\in \Re \\

1, & x\in \Re \quad x \notin A\\

\end{array}\right. $

Thank you for your time.