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Homework Help: Proof? (Real Analysis 1)

  1. Aug 28, 2010 #1
    1. The problem statement, all variables and given/known data

    See attachment

    2. Relevant equations

    3. The attempt at a solution

    It is not a homework. I am just reviewing for myself.
    This is the very first, basic problem of the first chapter of Real Analysis 1 by Bartle and Sherbert. Proofs of Real Analysis don't make any sense to me.

    Attached Files:

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  2. jcsd
  3. Aug 28, 2010 #2
    The "if and only if" denotes an equivalence relation between the two statements, i.e., we have [tex](A \subseteq B) \longleftrightarrow (A \cap B = A). [/tex] There are two steps necessary to proving an equivalence statement:

    1. Show [tex](A \subseteq B) \rightarrow (A \cap B = A). [/tex]
    2. Show [tex](A \cap B = A) \rightarrow (A \subseteq B). [/tex]

    For both parts, we begin by assuming the respective premise.
    For 1, we need to show that [tex] x \in (A \cap B) \rightarrow x \in A [/tex] and that [tex] x \in A \rightarrow x \in (A \cap B). [/tex]
    For 2, we just need to show that [tex] x \in A \rightarrow x \in B. [/tex]
    Do all this, and you have yourself a proof!
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