# Math Help - Natural Numbers

1. ## 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?

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

3. 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

4. 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 $x_0$.

5. Originally Posted by emakarov
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}

6. 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.

7. 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?

8. 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?