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Simple proof

  1. Sep 7, 2007 #1
    1. The problem statement, all variables and given/known data

    Trying to prove some of the basic laws in set theory, and would like any opinions on 1 of my proofs (eg hints on how can I improve it, is it even a valid proof). Thanks in advance.

    [tex](A \subseteq B \wedge B \subseteq C) \rightarrow (A \subseteq C)[/tex]

    2. Relevant equations

    3. The attempt at a solution

    1) [tex]\forall x \in A, x \in B$[/tex] - definition of a subset
    2) [tex]\forall x \in B, x \in C[/tex] - definition of a subset
    3) [tex]\forall x \in A, x \in C[/tex] - 1, 2
    4) [tex]A \subseteq C[/tex] - 3, definition of a subset
  2. jcsd
  3. Sep 7, 2007 #2


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    Your logic is correct.

    You could also use proof by contradiction:

    Suppose A [itex]\nsubseteq[/itex] C. Then there must be some a in A that is not in C. Since B [itex]\subseteq[/itex] C, a cannot be in B. This contradicts A [itex]\subseteq[/itex] B.
    Last edited: Sep 7, 2007
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