Results 1 to 7 of 7
Like Tree4Thanks
  • 1 Post By chiro
  • 1 Post By Plato
  • 1 Post By Plato
  • 1 Post By Plato

Thread: Help understanding this proof: E is a subset of f^(-1)(f(E))

  1. #1
    Junior Member
    Joined
    Sep 2011
    From
    NA
    Posts
    50

    Help understanding this proof: E is a subset of f^(-1)(f(E))

    \text{Let } f: X\rightarrow Y, \text{ where } X \text{ and } Y \text{ are sets.}

    \text{Prove that }E \subseteq f^{-1}(f(E)) \text{ } \forall E \subseteq X.

    PROOF:
    \text{Let } x \in E.
    \text{Then, }y = f(x) \in f(E), \text{ so } x \in f^{-1}(f(E)).
    \text{ Hence } E \subseteq f^{-1}(f(E))

    How do we know that y = f(x) \in f(E)?

    And how does that mean that x \in f^{-1}(f(E))?

    I guess I'm struggling with comprehending how a function is defined. We're using the book called Mathematical Analysis by Apostol and it's not an easy read for me.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    6,577
    Thanks
    1711

    Re: Help understanding this proof: E is a subset of f^(-1)(f(E))

    Hey amthomasjr.

    A function is defined as a mapping from one set to another where the mapping is one to one [often known as bijective].

    The "funny" e sign means "is an element of" which means if you have a collection of "things" then there is an element inside that collection that satisfies some "criterion".
    Thanks from amthomasjr
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,453
    Thanks
    2728
    Awards
    1

    Re: Help understanding this proof: E is a subset of f^(-1)(f(E))

    Quote Originally Posted by amthomasjr View Post
    \text{Let } f: X\rightarrow Y, \text{ where } X \text{ and } Y \text{ are sets.}
    \text{Prove that }E \subseteq f^{-1}(f(E)) \text{ } \forall E \subseteq X.
    PROOF:
    \text{Let } x \in E.
    \text{Then, }y = f(x) \in f(E), \text{ so } x \in f^{-1}(f(E)).
    \text{ Hence } E \subseteq f^{-1}(f(E))
    How do we know that y = f(x) \in f(E)?
    It is really knowing the definitions,
    If $f:X\to Y$ and $E\subset X$, then by definition $\large f(E)=\{f(x) :x\in E\}$
    $f:X\to Y$ and $H\subset Y$, then by definition $\large f^{-1}(H)=\{t\in X :f(t)\in H\}$
    Thanks from amthomasjr
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Sep 2011
    From
    NA
    Posts
    50

    Re: Help understanding this proof: E is a subset of f^(-1)(f(E))

    Quote Originally Posted by Plato View Post
    It is really knowing the definitions,
    If $f:X\to Y$ and $E\subset X$, then by definition $\large f(E)=\{f(x) :x\in E\}$
    $f:X\to Y$ and $H\subset Y$, then by definition $\large f^{-1}(H)=\{t\in X :f(t)\in H\}$
    So I just let f(E) = H since f(E) \subseteq Yand apply it to the second definition. That makes sense.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,453
    Thanks
    2728
    Awards
    1

    Re: Help understanding this proof: E is a subset of f^(-1)(f(E))

    Quote Originally Posted by amthomasjr View Post
    So I just let f(E) = H since f(E) \subseteq Yand apply it to the second definition.
    From these
    If $f:X\to Y$ and $E\subset X$, then by definition $\large f(E)=\{f(x) :x\in E\}$
    $f:X\to Y$ and $H\subset Y$, then by definition $\large f^{-1}(H)=\{t\in X :f(t)\in H\}$

    We know that $f(E)\subset Y$ thus $f^{-1}(f(E))\subset X$.

    If $t\in E$ then $f(t)\in f(E)$ so by definition $t\in f^{-1}(f(E))$ by definition.
    Thanks from amthomasjr
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Sep 2011
    From
    NA
    Posts
    50

    Re: Help understanding this proof: E is a subset of f^(-1)(f(E))

    Quote Originally Posted by Plato View Post
    From these
    If $t\in E$ then $f(t)\in f(E)$ so by definition $t\in f^{-1}(f(E))$ by definition.
    Suppose I was only given $f(t)\in f(E)$. Is deducing that t \in E valid? I know in some cases the converse of a conditional isn't logically equivalent.
    Last edited by amthomasjr; Sep 16th 2016 at 04:45 PM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,453
    Thanks
    2728
    Awards
    1

    Re: Help understanding this proof: E is a subset of f^(-1)(f(E))

    Quote Originally Posted by amthomasjr View Post
    Suppose I was only given $f(t)\in f(E)$. Is deducing that t \in E valid?
    No That is not valid.

    BUT in the OP you were told to show that $\large E\subset f^{-1}(f(E)).$
    Thanks from amthomasjr
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Really need some help with a subset proof!
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Oct 26th 2012, 09:07 AM
  2. Need help with a subset proof
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: Apr 22nd 2012, 06:08 PM
  3. [SOLVED] Proof that if A is a subset of P(A), P(A) is a subset of P(P(A))
    Posted in the Discrete Math Forum
    Replies: 12
    Last Post: May 23rd 2011, 05:26 PM
  4. Proof on Openness of a Subset and a Function of This Subset
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Oct 24th 2010, 10:04 PM
  5. proof about a subset
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Oct 12th 2009, 10:21 AM

Search Tags


/mathhelpforum @mathhelpforum