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Thread: set theory/logical proof

  1. #1
    Junior Member
    Nov 2009

    set theory/logical proof

    Prove that for all natural numbers x and y, if x is not equal to y, the S(x) is not equal to S(y).

    Im thinking it might make more sense to prove that if x=y that gives that S(x)=S(y)

    but im having difficulties getting started because it seems obvious.
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  2. #2
    MHF Contributor
    Oct 2009
    As in the other thread, this depends on the content of the course/textbook. If this is about Peano arithmetic, then x = y -> S x = S y holds not only for S but for any unary functional symbol. This is an instance of one of the axioms for equality (and Peano arithmetic is by definition a first-order theory with equality).

    On the other hand, x # y -> S x # S y, which is equivalent to S x = S y -> x = y says that S is injective. Other functional symbols may be interpreted by non-injective functions. Injectivity of S is also one of the Peano axioms.
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