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Math Help - proof of axioms

  1. #1
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    proof of axioms

    given that set B is ccontained in a set A, show that the probability of A is greater than or equal to that of B
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    Short proof: All the elements in B are also in A.

    Long proof:

    \displaystyle B \subseteq A

    \displaystyle n(B) \leq n(A)

    \displaystyle \frac{n(B)}{n(\varepsilon)} \leq \frac{n(A)}{n(\varepsilon)}

    \displaystyle Pr(B) \leq Pr(A).
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    Quote Originally Posted by Prove It View Post
    Short proof: All the elements in B are also in A.

    Long proof:

    \displaystyle B \subseteq A

    \displaystyle n(B) \leq n(A)

    \displaystyle \frac{n(B)}{n(\varepsilon)} \leq \frac{n(A)}{n(\varepsilon)}

    \displaystyle Pr(B) \leq Pr(A).
    What is n(.) and what rules of proof are in force here?

    Also re thread title: You can't proove axioms (unless there is redundancy in the set of axioms but we do try to avoid that).

    Come to think of it do you not have to assume that both probabilities exist?

    CB
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    Quote Originally Posted by CaptainBlack View Post
    What is n(.) and what rules of proof are in force here?

    Also re thread title: You can't proove axioms (unless there is redundancy in the set of axioms but we do try to avoid that).

    Come to think of it do you not have to assume that both probabilities exist?

    CB
    The n(A) notation stands for number of elements in A.
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    Quote Originally Posted by Prove It View Post
    The n(A) notation stands for number of elements in A.
    Are you then assuming finite sets and equally likely cases, I'm not sure your argument works even with countably infinite sets.

    You also leave too much notation undefined.

    CB
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    Quote Originally Posted by CaptainBlack View Post
    Are you then assuming finite sets and equally likely cases, I'm not sure your argument works even with countably infinite sets.

    You also leave too much notation undefined.

    CB
    I disagree, this is standard notation - even year 8 students should be aware of the notation of \displaystyle \varepsilon to represent the universal set, \displaystyle n(A) to represent the number of elements in set A, and that \displaystyle Pr(A) = \frac{n(A)}{n(\varepsilon)}. Even if they are infinite, the same logic holds if you were to draw a Venn Diagram of the situation.
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    Quote Originally Posted by Prove It View Post
    I disagree, this is standard notation - even year 8 students should be aware of the notation of \displaystyle \varepsilon to represent the universal set,
    Ha ha ha ... you jest


    \displaystyle n(A) to represent the number of elements in set A,
    Cardinality? That is not the notation I know.

    and that \displaystyle Pr(A) = \frac{n(A)}{n(\varepsilon)}. Even if they are infinite, the same logic holds if you were to draw a Venn Diagram of the situation.
    Let your \varepsilon be the unit disk, let A be some disk contained in \varepsilon, now what does your notation mean?



    CB
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    Quote Originally Posted by omoboye View Post
    given that set B is ccontained in a set A, show that the probability of A is greater than or equal to that of B
    The title did say “from the axioms”
    Recall that \left( {\forall C} \right)\left[ {P(C) \geqslant 0} \right] and C \cap D = \emptyset \; \Rightarrow \;P(C \cup D) = P(C) + P(D).

    Use those. A = B \cup \left( {A\backslash B} \right), therefore P(A) = P(B) + P\left( {A\backslash B} \right) \geqslant P(B).
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