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Proove that 1 is not equal to 0??

  • 11-04-2008 11:14AM
    #1
    sd123
    Closed Accounts Posts: 534 ✭✭✭


    Someone challenged me to do it but ive no idea where to even start, is it true that it's one of the first things they teach you in college maths... any help on getting me started?


Comments

  • #2
    Thomas_S_Hunterson
    Closed Accounts Posts: 6,151 ✭✭✭Thomas_S_Hunterson


    I don't know, we certainly haven't looked at it yet.

    You could try a proof by contradiction, although I'm not sure if it'd be totally rigorous.

    Surely they aren't equal by definition


  • #3
    Fremen
    Registered Users, Registered Users 2 Posts: 2,481 ✭✭✭Fremen


    This depends on which set of axioms you choose in order to define your number system.
    It should be fairly trivial to derive a contradiction with a sensible choice of axioms. For instance, working with the integers, assume there exist two integers M and N with (N < M).
    Assume also that multiplication distributes over addition, i.e.

    a(b + c) = ab + ac

    then set a = 1 , b = N, c = 0

    1(N + 0) = N + 1 (setting the zero in the brackets to 1, by assumption, and using the fact that 1 is the multiplicative identity)

    but

    1(N + 0) = 1(N) using the fact that zero is the additive identity.

    then N + 1 = N

    Repeating this argument on the left hand side, we arrive at

    M = N which is a contradiction

    (Edit: I just noticed that you don't even need associativity, you juat add 0 to one side and 1 to the other. Thus, if there exist two distinct integers, then 1 is not equal to zero!)


  • #4
    Rattlehead_ie
    Registered Users, Registered Users 2 Posts: 1,893 ✭✭✭Rattlehead_ie


    Does Binary not prove this?
    1 and 0 are completely different when it comes to anything binary based?


  • #5
    Fremen
    Registered Users, Registered Users 2 Posts: 2,481 ✭✭✭Fremen


    Rattlehead_ie wrote: »
    Does Binary not prove this?
    1 and 0 are completely different when it comes to anything binary based?

    Binary is just a system for representing numbers. The problem is more abstract than that.

    You could rephrase it as:
    given a number X such that
    AX = XA = A for all A,

    and a number Y such that
    A + Y = Y + A = A, for all A

    prove X is not equal to Y


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