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Thread: Can someone help me prove this Set Theory basic rule?

  1. #1
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    Can someone help me prove this Set Theory basic rule?

    https://imgur.com/a/wgMAOOh

    I was given the constants, etc. but I am confused on the last part:

    How does A and B = A and B and C + A and B and C' (complement of C)?

    Is it because C and C' cancel out each other?
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  2. #2
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    Re: Can someone help me prove this Set Theory basic rule?

    $\begin{align*}

    &(A \cap B \cap C) \cup (A \cap B \cap C') = \\ \\

    &(A \cap B) \cap (C \cup C') = \\ \\

    &(A \cap B) \cap U = \\ \\

    &(A \cap B)

    \end{align*}$
    Thanks from math951
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  3. #3
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    Re: Can someone help me prove this Set Theory basic rule?

    Quote Originally Posted by math951 View Post
    https://imgur.com/a/wgMAOOh
    I was given the constants, etc. but I am confused on the last part:
    How does A and B = A and B and C + A and B and C' (complement of C)?
    Is it because C and C' cancel out each other?
    In the image I assume that $n(C)$ is the count in set $C$.
    So that
    $ \begin{align*}(A\cap B\cap C)\cup(A\cap B\cap C')&=(A\cap B)\cap(C\cup C')\\&=(A\cap B)\cap(\mathcal{U})\\&=(A\cap B) \end{align*}$

    $\mathcal{U}$ is the universe.

    I have no idea what the rest of that image means (is about).
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    Re: Can someone help me prove this Set Theory basic rule?

    I have to be honest, I still am not understanding this. How does A and B and C or A and B and C' = A and B and C or C'. Is this part of Demorgan's law?

    Here is question: https://imgur.com/a/h3xPgP0


    Here is solution: https://imgur.com/a/rJSGwmd
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  5. #5
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    Re: Can someone help me prove this Set Theory basic rule?

    Pretty sure I have to master set theory before I can move on to probability concepts.
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    Re: Can someone help me prove this Set Theory basic rule?

    Quote Originally Posted by math951 View Post
    Pretty sure I have to master set theory before I can move on to probability concepts.
    I taught probability theory to undergraduates for thirty+ years. I began each term sayings "If you are not well grounded in set theory please go and drop this class from your schedule for this term, signup for foundations of mathematics".
    For sets $A,~B,~\&~C$
    $A\subseteq A\cup B\\ A\cap B\subseteq A\\ A\cap A'=\emptyset\\ A\cup A'=\mathcal{U}$

    Now $n(A\cup B)=n(A)+n(B)-n(A\cap B)$
    so if $A\cap B=\emptyset$ then $n(A\cup B)=n(A)+n(B)$

    This gives some idea why you will need a grounding in sets.
    Thanks from math951 and topsquark
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  7. #7
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    Re: Can someone help me prove this Set Theory basic rule?

    Gotcha. I actually am studying for Exam P right now (actuary). in the fall quarter, I will be taking Math 170A at UCLA, which is: Probability and stochastic processes are used to create and analyze models in a broad range of fields, including statistics, economics, finance, engineering, biology and physics. Mathematics 170AB and 171 are designed to give a firm foundation in this area for students who will work and/or do graduate work in one of these fields. They also provide an excellent background for graduate work in probability and related areas of mathemati

    Do you have any recommended sources to study set theory? Preferably free? I am watching lectures on youtube right now of this guy for it: https://www.youtube.com/watch?v=qgiz...zwxR6mPyTyc6ne
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