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Math Help - Checking wether a 2 variable 2 valued function is bijective

  1. #1
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    Checking wether a 2 variable 2 valued function is bijective

    Hi everybody

    Could anyone please tell me how to check or prove if a function from [0,1]^2 to [0,1]^2 is a bijection. It easy from [0,1] to [0,1], but not so obvious in the 2 dimensional case. The type of function I am thinking of is continuous but not necessarly differentiable (e.g I don't want to have to use partial derivatives)

    thanks
    Last edited by SkanderH; January 22nd 2006 at 08:10 PM.
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  2. #2
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    What is ^2? What type of function is that?

    Use the mathematical meaning of bijective.
    I do not want to get in to the whole definition of a function from a set-theoretic viewpoint, but the basic thing you need to show 2 things.

    Step 1:One to One Show that f(x_1,y_1)=f(x_2,y_2) if and only if (x_1,y_1)=(x_2,y_2)

    Step 2:Surjective Show that for any r\in R (any real number) the equation f(x,y)=r has a solution.

    Sometimes showing part 2 is more difficult, because you prove existence for that part. Sometime what you can do is prove that the function is countinous, then by the intermediate value theorem it must have a solution.

    Of course, if you need to show that a function is bijective in a disk then do the same steps but only limited to that disk.
    Last edited by ThePerfectHacker; January 22nd 2006 at 08:18 PM.
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  3. #3
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    Thanks for the reply
    by [0,1]^2, I meant [0,1] to the power of 2, the unit square.
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  4. #4
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    Any specific problem?
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  5. #5
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    Yes

    I am trying to find a Fuzzy logic version of reversible boolean logic gates like the Toffoli gate or the Fredkin gate.

    I am first trying the most basic reversibel gate, the C-not gate, which in boolean maps (a,b) to (a,a xor b), but all T-norm and S-norms I used produce irreversible gates.
    The fuzzy version should map from the unit square to the unit square again. I don't if the problem is with my choice of T-noms, or is the problem inherent in the characteristic of C-not.

    Here are some links:
    http://en.wikipedia.org/wiki/Reversible_computing
    http://en.wikipedia.org/wiki/Toffoli_gate
    http://en.wikipedia.org/wiki/Fredkin_Gate
    http://en.wikipedia.org/wiki/T-norm
    Last edited by SkanderH; January 23rd 2006 at 06:04 PM.
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  6. #6
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    Quote Originally Posted by SkanderH
    Yes

    I am trying to find a Fuzzy logic version of reversible boolean logic gates like the Toffoli gate or the Fredkin gate.

    I am first trying the most basic reversibel gate, the C-not gate, which in boolean maps (a,b) to (a,a xor b), but all T-norm and S-norms I used produce irreversible gates.
    The fuzzy version should map from the unit square to the unit square again. I don't if the problem is with my choice of T-noms, or is the problem inherent in the characteristic of C-not.

    Here are some links:
    http://en.wikipedia.org/wiki/Reversible_computing
    http://en.wikipedia.org/wiki/Toffoli_gate
    http://en.wikipedia.org/wiki/Fredkin_Gate
    http://en.wikipedia.org/wiki/T-norm
    I am afraid I cannot help you with that I have never studied that.
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