Thread: 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?

Please post the relevant details, in particular definitions. There are probably dozens of equivalent ways to define natural numbers, subtraction and >.

Originally Posted by emakarov

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

I don't understand the definition . The variable is not bound (by a quantifier), so it is not clear whether is true or false for each particular .

Originally Posted by emakarov

I don't understand the definition . The variable is not bound (by a quantifier), so it is not clear whether is true or false for each particular .

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}

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.

Originally Posted by emakarov

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?

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?