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Math Help - Proof that 1 is in the Natural Numbers

  1. #1
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    Proof that 1 is in the Natural Numbers

    1 is an element of the Natural Numbers.

    Hey all, I would really appreciate some help with the above proof. Thanks a bunch!
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  2. #2
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    The answer to this question, and others you posted in this forum recently, depends on the context. If you need a secondary-school proof, then this is the wrong forum because Number Theory is an advance university subject. Even pre-university forums deal mostly with high-school math, whereas the fact that 1 is a natural number is known in elementary school.

    On the other hand, if you need a formal proof, then you need to specify the definitions and axioms you are using, such as the definition of natural numbers in set theory, ring axioms, Peano axioms, etc.
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  3. #3
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    Quote Originally Posted by jstarks44444 View Post
    1 is an element of the Natural Numbers.

    Hey all, I would really appreciate some help with the above proof. Thanks a bunch!
    Tell us what definition of the Naturals you are using.

    In Peano arithmetic if we start with 0, and the successor operation s and define n to be a natural if it is either 0 or the successor of a natural number, then as "1" is the name of s0 it is trivially a natural number.

    If we start Peano arithmetic with 1 and the successor operation then "1" is by definition a natural number.

    Simular arguments apply to the set theoretic construction on the Naturals.

    CB
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  4. #4
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    The Naturals in this context is defined by the following:

    There exists a subset N in Z (integers) with the following properties:

    (i) If m,n are elements of N then m+n is an element of N
    (ii) If m,n are elements of N then mn is an element of N
    (iii) 0 is not an element of N
    (iv) For every m that is an element of Z, we have m is an element of N or m=0 or -m is an element of N

    This proof needs to be proved by contradiction by the way.
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  5. #5
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    Quote Originally Posted by jstarks44444 View Post
    There exists a subset N in Z (integers) with the following properties:
    (i) If m,n are elements of N then m+n is an element of N
    (ii) If m,n are elements of N then mn is an element of N
    (iii) 0 is not an element of N
    (iv) For every m that is an element of Z, we have m is an element of N or m=0 or -m is an element of N
    This proof needs to be proved by contradiction by the way.
    Suppose that 1\notin \mathbb{N}.
    We know that 1\in \mathbb{Z} so then either 1=0 or -1\in \mathbb{N} (iv).
    From the integer properties you should have that 1\not= 0.
    That leaves -1\in \mathbb{N}.
    Can you finish? [hint: (ii)]
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  6. #6
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    (-1)(-1) leads to a contradiction! Thanks a lot for the help!
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