Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    EnumaElish

    User Avatar
    Science Advisor
    Homework Helper

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook