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Math Help - Natural Numbers

  1. #1
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    Natural Numbers

    If x and y are natural numbers and x > y, then x-y is also a natural number.

    how to prove this by induction?

    is it becuase x-1 is a natural number,

    and then for a number k, x - (k+1) = x-k-1=(x-k)-1 is also a natural number?
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  2. #2
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    Please post the relevant details, in particular definitions. There are probably dozens of equivalent ways to define natural numbers, subtraction and >.
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  3. #3
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    Quote Originally Posted by emakarov View Post
    Please post the relevant details, in particular definitions. There are probably dozens of equivalent ways to define natural numbers, subtraction and >.
    The reason my book states to show is that the set
    S= {x E N| y E N or x-y E N} is inductive

    I have no idea on how to work on the sets. The only thing I can think of is what I stated in the question
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  4. #4
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    I don't understand the definition S=\{x\in\mathbb{N}\mid y\in\mathbb{N}\lor x-y\in\mathbb{N}\}. The variable y is not bound (by a quantifier), so it is not clear whether y\in\mathbb{N}\lor x_0-y\in\mathbb{N} is true or false for each particular x_0.
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  5. #5
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    Quote Originally Posted by emakarov View Post
    I don't understand the definition S=\{x\in\mathbb{N}\mid y\in\mathbb{N}\lor x-y\in\mathbb{N}\}. The variable y is not bound (by a quantifier), so it is not clear whether y\in\mathbb{N}\lor x_0-y\in\mathbb{N} is true or false for each particular x_0.
    Sorry, I wrote the wrong thing. It is supposed to be
    S(y) = { y E N | x E N, if x>y, x-y E N}
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  6. #6
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    If x and y are natural numbers and x > y, then x-y is also a natural number.
    I guess, a simple answer is to do induction on x with the base case on x=y+1. Then if for x there is a number z (= x-y) such that y+z = x, then a number with the same property exists for x+1.
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  7. #7
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    Quote Originally Posted by emakarov View Post
    I guess, a simple answer is to do induction on x with the base case on x=y+1. Then if for x there is a number z (= x-y) such that y+z = x, then a number with the same property exists for x+1.
    I have to do induction on y, so is it similar to what you have stated here?
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  8. #8
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    S is the set of all natural numbers, y, such that "if x is a natural number and x> y then x-y is a natural number".

    Your induction must start with y= 1. Show that if x is a natural number and x> 1 then x- 1 is a natural number.

    But you still haven't done what emakarov asked you to do in his first reponse: what is your definition of ">" and what is your definition of "x- y" for natural numbers?
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