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Thread: Indicator function properites

  1. #1
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    Indicator function properites

    I just wanted to make sure my proofs are correct.
    Given the indicator function.

    $\displaystyle 1_A = \left\{ \begin{array}{rcl}
    1 & \mbox{for} & x \in A \\
    0 & \mbox{for} & x \notin A
    \end{array}\right.$

    show that:
    i) $\displaystyle 1_{A \cap B} = 1_A \cdot 1_B$:

    Proof:
    $\displaystyle 1_{A \cap B} = \left\{ \begin{array}{rcl}
    1 & \mbox{for} & x \in A\cap B \\
    0 & \mbox{for} & x \notin A\cap B
    \end{array}\right.$ which implies that $\displaystyle 1_{A \cap B} = 1_A \cdot \left\{ \begin{array}{rcl}
    1 & \mbox{for} & x \in B \\
    0 & \mbox{for} & x \notin B
    \end{array}\right. = 1_A \cdot 1_B$


    ii) $\displaystyle 1_{A \cup B} = 1_A + 1_B - 1_A \cdot 1_B$

    Proof:
    $\displaystyle 1_{A \cup B} = \left\{ \begin{array}{rcl}
    1 & \mbox{for} & x \in A\cup B \\
    0 & \mbox{for} & x \notin A\cup B
    \end{array}\right.$ now $\displaystyle A\cup B$ can be written as two disjoint sets so we have

    $\displaystyle A\cup B \Rightarrow A\cup(B-A\cap B) $ so we have :

    $\displaystyle 1_{A \cup B} = \left\{ \begin{array}{rcl}
    1 & \mbox{for} & x \in A\cup(B-A\cap B) \\
    0 & \mbox{for} & x \notin A\cup(B-A\cap B)
    \end{array}\right. = 1_A +\left\{ \begin{array}{rcl}
    1 & \mbox{for} & x \in B -A\cap B \\
    0 & \mbox{for} & x \notin B -A\cap B
    \end{array}\right. = 1_A+1_B-1_A\cdot 1_B$ where the terms are expanded out.

    iii) $\displaystyle 1_{A \cap B} =\min(1_A,1_B)$

    I have no idea
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  2. #2
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    $\displaystyle \min\{1_A,1_B\}=0$ if and only if $\displaystyle 1_A=0\text{ or }1_B=0$.

    Does that work?

    P.S. Here is another ‘fun’ one for symmetric difference.

    $\displaystyle 1_{A\Delta B}=1_A+1_B-2\cdot 1_A\cdot 1_B$.
    Last edited by Plato; Oct 8th 2010 at 02:19 PM.
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