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Math Help - one one and onto

  1. #1
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    one one and onto

    is monotonicity is sufficient condition for proving one one .do every function must satisfy this condition.
    also what is best method to prove onto
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by ayushdadhwal View Post
    is monotonicity is sufficient condition for proving one one .do every function must satisfy this condition.
    also what is best method to prove onto
    If by monotone increasing you mean (the opposite for decreasing) the important part is the strict inequality.

    x_1 < x_2 \implies f(x_1) < f(x_2) then this is sufficient to prove 1-1.

    Suppose f is monotone. Now suppose by way of contradiction that f is not 1-1 then there exists x_1 \ne x_2 such that f(x_1)=f(x_2) without loss of generality assume that x_1 < x_2 then by the monotone property we have that

    f(x_2) = f(x_1) < f(x_2)=f(x_1) but this is a contradiction!

    Onto is false consider any piecewise increasing function with jump discontinuities.
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  3. #3
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    sir good evening
    sir suppose i choose a function with restricted domain{1,2,3,4) such that f(1)=2 f(2)=1 f(3)=6 f(4)=-3 (ie point graph will be formed)
    now this function is one one but not neither decreasing nor increasing .little bit doubt sir please clear
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ayushdadhwal View Post
    sir good evening
    sir suppose i choose a function with restricted domain{1,2,3,4) such that f(1)=2 f(2)=1 f(3)=6 f(4)=-3 (ie point graph will be formed)
    now this function is one one but not neither decreasing nor increasing .little bit doubt sir please clear
    I think there is a slight bit of confusion going on here. It's true, for example, that for continuous f:\mathbb{R}\to\mathbb{R} one has that strict monotonicity is equivalent to injectiveness.
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