Results 1 to 2 of 2

Thread: Equality of sets

  1. #1
    Senior Member oldguynewstudent's Avatar
    Joined
    Oct 2009
    From
    St. Louis Area
    Posts
    255

    Equality of sets

    I have a final tonight and I forgot how to do this:

    Let f: X \longrightarrow Y be a function. Show that

    (a) f(S \cap T)) \subset f(S) \cap f(T)

    (b) If f is 1-1, then show that the above inclusion is actually an equality of sets.

    (a) f(S \cap T) \equiv { y | y = f(x), x \epsilon S and x \epsilon T} \equiv {y | y = f(x), x \epsilon S} and {y | y = f(x), x \epsilon T} \subseteq f(S) \cap f(T)

    (b) Is the following correct?

    If f 1-1 then f( x_1) = f( x_2) \longrightarrow x_1 = x_2

    We have already shown that f(S \cap T)) \subset f(S) \cap f(T). It is left to show that f(S) \cap f(T) \subset f(S \cap T)).

    If f(x) \epsilon f(S) \cap f(T), then f(x) \epsilon f(S) and f(x) \epsilon f(T). Because f is 1-1, x \epsilon S and x \epsilon T. This means x \epsilon S \cap T. Again because f is 1-1, f(x) \epsilon f(S \cap T). This proves each is a subset of the other so they are equivalent.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    20,281
    Thanks
    2217
    Awards
    1
    Quote Originally Posted by oldguynewstudent View Post
    We have already shown that f(S \cap T)) \subset f(S) \cap f(T). It is left to show that f(S) \cap f(T) \subset f(S \cap T)).

    If f(x) \epsilon f(S) \cap f(T), then f(x) \epsilon f(S) and f(x) \epsilon f(T). Because f is 1-1, x \epsilon S and x \epsilon T. This means x \epsilon S \cap T. Again because f is 1-1, f(x) \epsilon f(S \cap T). This proves each is a subset of the other so they are equivalent.
    The part in red has a logical error but the idea is correct.
    \begin{gathered}<br />
  z \in f(S) \cap f(T) \hfill \\<br />
  z \in f(S)\;\& \;z \in f(T) \hfill \\<br />
  \left( {\exists s \in S} \right)\left[ {f(s) = z} \right]\;\& \;\left( {\exists t \in T} \right)\left[ {f(t) = z} \right] \hfill \\<br />
  z = f(s) = f(t)\; \Rightarrow \;s = t \hfill \\<br />
  \; \Rightarrow \;z \in f(S \cap T) \hfill \\ <br />
\end{gathered}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Logic and Sets - Equality
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Dec 7th 2011, 07:07 AM
  2. [SOLVED] Equality of two sets (using Boolean algebra)
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: Apr 2nd 2011, 11:25 AM
  3. Prove the following equality of sets
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Jul 30th 2010, 08:45 PM
  4. Equality of Sets
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Oct 8th 2009, 04:57 PM
  5. Cartesian Products: Proof of Equality of Sets
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: May 9th 2006, 02:46 PM

Search Tags


/mathhelpforum @mathhelpforum