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

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

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

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

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

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

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

    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.
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  2. #2
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    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
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    Re: Help understanding this proof: E is a subset of f^(-1)(f(E))

    Quote Originally Posted by amthomasjr View Post
    $\displaystyle \text{Let } f: X\rightarrow Y, \text{ where } X \text{ and } Y \text{ are sets.}$
    $\displaystyle \text{Prove that }E \subseteq f^{-1}(f(E)) \text{ } \forall E \subseteq X.$
    PROOF:
    $\displaystyle \text{Let } x \in E.$
    $\displaystyle \text{Then, }y = f(x) \in f(E), \text{ so } x \in f^{-1}(f(E)).$
    $\displaystyle \text{ Hence } E \subseteq f^{-1}(f(E))$
    How do we know that $\displaystyle y = f(x) \in f(E)$?
    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
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    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 $\displaystyle f(E) = H$ since $\displaystyle f(E) \subseteq Y$and apply it to the second definition. That makes sense.
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    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 $\displaystyle f(E) = H$ since $\displaystyle f(E) \subseteq Y$and 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
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    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 $\displaystyle $f(t)\in f(E)$$. Is deducing that $\displaystyle t \in E$ valid? I know in some cases the converse of a conditional isn't logically equivalent.
    Last edited by amthomasjr; Sep 16th 2016 at 03:45 PM.
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    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 $\displaystyle $f(t)\in f(E)$$. Is deducing that $\displaystyle t \in E$ valid?
    No That is not valid.

    BUT in the OP you were told to show that $\large E\subset f^{-1}(f(E)).$
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