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Math Help - Topology: How to formally write proofs?

  1. #1
    Newbie MadMikey's Avatar
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    Topology: How to formally write proofs?

    Hey there! In doing practice exercises for topology, I find that I can reason through them easily enough to find the correct answer. The problem is that I rely predominantly upon verbal logic in order to make the proofs work. How does one go about proofs using only the math notation?

    A simple example: Prove that A\in 2^A

    Another simple example: Prove that A \subset B \iff A \cup\B = B

    thanks for your time.
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  2. #2
    Member ModusPonens's Avatar
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    Re: Topology: How to formally write proofs?

    You have to use english, or else you would have to resort to symbolic derivations in some logic. See the examples of proofs in a topology book and you'll get the style.
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    Re: Topology: How to formally write proofs?

    Quote Originally Posted by MadMikey View Post
    Hey there! In doing practice exercises for topology, I find that I can reason through them easily enough to find the correct answer. The problem is that I rely predominantly upon verbal logic in order to make the proofs work. How does one go about proofs using only the math notation?

    A simple example: Prove that A\in 2^A
    The standard way to prove that x\in U is to show that x meets the conditions given in the definition of U. What is the definition of 2^A?

    Another simple example: Prove that A \subset B \iff A \cup B = B
    The standard way to prove " X= Y" is to prove both X\subset of Y and Y\subset of X. And the standard way to prove " X\subset Y" is to start "if x\in X" and use the definitions of both X and Y to prove that " x\in Y".

    For example, to prove "if A\cup B= B then A\subset B" start by saying "if x \in A\cup B" and argue that, then, x\in A\cup B and so, because A\cup B= B, x\in B. Since every member of A is a member of B, A\subset B. I'll leave the other way (which is a little harder!) to you.

    thanks for your time.
    (Your B in A\cup\B did not show up because you had "\" in front of the B and Latex does not recognize "\B".)
    Last edited by HallsofIvy; December 5th 2012 at 06:48 AM.
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    Re: Topology: How to formally write proofs?

    something to help you proving:

    A ⊆ B implies AUB = B.

    the way we show two sets are equal is to demonstrate they contain the same elements. so if we start with A ⊆ B, we need to show that everything in AUB is in B, and that everything in B is in AUB.

    now everything in B is always in AUB (by the definition of union). so the "hard part" will be showing that everything in AUB is in B.

    you'd start like so. suppose x is in AUB = {x in T : x is in A or x is in B} (here, T is "some set" that A and B both belong to, our "universe of discourse").

    so either x is in A, or x is in B.

    if x is in B, then certainly x is in B.

    on the other hand, if x is in A, then.....(you should use something about the relationship between A and B here, now)
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