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Math Help - a question about set relation between functions

  1. #1
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    Mumbai, India
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    a question about set relation between functions

    hi

    I have few doubts about one set theorem which I proved.

    Suppose that f:X\rightarrow Y with A\subseteq X
    and B\subseteq Y . I had to prove that

    A\subseteq f^{-1}(f(A))

    I did prove that but I have some questions. Shouldn't there be an equality

    A= f^{-1}(f(A)) because when we talk about the inverse functions, we talk about one-to-one functions. So I can't think of situation
    where

    \textstyle A\subset f^{-1}(f(A))

    Can you ? and question related to latex typesetting. how do I reduce font size ?

    thanks
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  2. #2
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    Quote Originally Posted by issacnewton View Post
    hi

    I have few doubts about one set theorem which I proved.

    Suppose that f:X\rightarrow Y with A\subseteq X
    and B\subseteq Y . I had to prove that

    A\subseteq f^{-1}(f(A))

    I did prove that but I have some questions. Shouldn't there be an equality

    A= f^{-1}(f(A)) because when we talk about the inverse functions, we talk about one-to-one functions. So I can't think of situation
    where

    \textstyle A\subset f^{-1}(f(A))

    Can you ? and question related to latex typesetting. how do I reduce font size ?

    thanks


    In this case f^{-1}(A) does not denote the inverse function of f applying on a set A, but the

    inverse image of A under ANY function f. Two very different concepts. For example

    f:\mathbb{R}\rightarrow \mathbb{R}_+\,,\,\,f(x):=x^2\,,\,\,A=[-1,1)\Longrightarrow f(A)=[0,1]\Longrightarrow A\subset f^{-1}(f(A))=[-1,1]

    Tonio
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  3. #3
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    Mumbai, India
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    thanks tonio

    that makes sense
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