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Math Help - A problem with sets

  1. #1
    Member Mollier's Avatar
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    A problem with sets

    Hi.

    problem:

    Show that if f:A\rightarrow B and E,F are subsets of A, then
    f(E\cup F)=f(E)\cup f(F) and f(E\cap F)\subseteq  f(E) \cap f(F).

    ---------------------------------------------------------------------------------------------------------------

    attempt:


    First I try to show that f(E\cup F)\subset f(E) \cup f(F) and then that f(E) \cup f(F) \subset f(E\cup F).

    y\in f(E\cup F) \Rightarrow f^{-1}(y)\in E\cup F \Rightarrow f^{-1}(y)\in E \; or \; f^{-1}(y)\in F.
    f(f^{-1}(y))=y\in f(E\cup F) and so f(E\cup F)\subset f(E) \cup f(F).

    y\in f(E)\cup f(F) \Rightarrow f^{-1}(y)\in E or f^{-1}(y)\in F \Rightarrow f^{-1}(y)\in E\cup F.
    f(f^{-1}(y))=y \in f(E\cup F) and so f(E) \cup f(F) \subset f(E\cup F).

    A friend of mine pointed out that I could show this in the following manner:

    <br />
\begin{aligned}<br />
f(E\cup F) =&\; \{f(x): x\in E \; or \; x\in F\}\\<br />
              =&\; \{f(x): x\in E\} \cup \{f(x): x\in F\}\\<br />
              =&\; f(E) \cup f(F)<br />
\end{aligned}<br />


    y\in f(E\cap F) \Rightarrow f^{-1}(y)\in E\cap F \Rightarrow f^{-1}(y)\in E \; and \; f^{-1}(y)\in F.
    Then f(f^{-1}(y))\in f(E)\cap f(F) \Rightarrow f(E\cap F)\subset f(E) \cap f(F).

    I do not know how to continue. If y\in f(E) \cap f(F), what then? Should I be doing all of this differently?

    Thank you.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Mollier View Post
    Hi.

    problem:

    Show that if f:A\rightarrow B and E,F are subsets of A, then
    f(E\cup F)=f(E)\cup f(F) and f(E\cap F)\subseteq  f(E) \cap f(F).

    ---------------------------------------------------------------------------------------------------------------

    attempt:


    First I try to show that f(E\cup F)\subset f(E) \cup f(F) and then that f(E) \cup f(F) \subset f(E\cup F).

    y\in f(E\cup F) \Rightarrow f^{-1}(y)\in E\cup F \Rightarrow f^{-1}(y)\in E \; or \; f^{-1}(y)\in F.
    f(f^{-1}(y))=y\in f(E\cup F) and so f(E\cup F)\subset f(E) \cup f(F).
    I don't see anywhere in the hypotheses that says f is invertible so you should not be writing " f^{-1}(y)". Write, rather, "if y\in f(E\cup F) then there exist x in E\cup F such that f(x)= y. Since x\in E\cup F, either x\in E or x\in F.

    Case 1: if x\in E then y\in f(E).

    Case 2: if x\in F then y\in f(F).

    y\in f(E)\cup f(F) \Rightarrow f^{-1}(y)\in E or f^{-1}(y)\in F \Rightarrow f^{-1}(y)\in E\cup F.
    f(f^{-1}(y))=y \in f(E\cup F) and so f(E) \cup f(F) \subset f(E\cup F).

    A friend of mine pointed out that I could show this in the following manner:

    <br />
\begin{aligned}<br />
f(E\cup F) =&\; \{f(x): x\in E \; or \; x\in F\}\\<br />
              =&\; \{f(x): x\in E\} \cup \{f(x): x\in F\}\\<br />
              =&\; f(E) \cup f(F)<br />
\end{aligned}<br />
    So your friend was essentially telling you the same thing I just did!


    y\in f(E\cap F) \Rightarrow f^{-1}(y)\in E\cap F \Rightarrow f^{-1}(y)\in E \; and \; f^{-1}(y)\in F.
    Then f(f^{-1}(y))\in f(E)\cap f(F) \Rightarrow f(E\cap F)\subset f(E) \cap f(F).

    I do not know how to continue. If y\in f(E) \cap f(F), what then? Should I be doing all of this differently?

    Thank you.
    If y\in f(E\cap F) then there exist x in E\cap F such that f(x)= E. Since x\in E\cap F, then x\in E so y= f(x)\in f(E) and x\in F so y= f(x)\in f(F). Therefore y\in f(E)\cap f(F).
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  3. #3
    Member Mollier's Avatar
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    That was a crystal clear explanation, thank you very much!
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