# Cancellation property of addition of natural numbers

#### zcdfhn

I have to prove that for all k,m,n $$\in$$ $$N$$ that if m+k = n+k, then m=n.

The problem mentions that I must prove this by induction.

I did the base case k = 0: If m+0 = n+0, by identity m=n.

Then I attempt to show that m+1 = n+1 implies m=n, but I am stuck, I don't see how induction can be used to prove this and then next predicate that m+k = n+k implies m=n.

Any help would be greatly appreciated, and thanks in advance.

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#### berkeman

Mentor
I have to prove that for all k,m,n $$\in$$ $$N$$ that if m+k = n+k, then m=n.

The problem mentions that I must prove this by induction.

I did the base case k = 0: If m+0 = n+0, by identity m=n.

Then I attempt to show that m+1 = n+1 implies m=n, but I am stuck, I don't see how induction can be used to prove this and then next predicate that m+k = n+k implies m=n.

Any help would be greatly appreciated, and thanks in advance.
"show that m+1 = n+1 implies m=n"

Can you subtract 1 from both sides? Not sure if that's allowed in this step of the proof. It's less general than subtracting k from both sides of the original question, but not by much...

#### Hurkyl

Staff Emeritus
Gold Member
The problem mentions that I must prove this by induction.

I did the base case k = 0:
The problem didn't say you had to induct on k, though.

As an aside... doesn't
m+k = n+k implies m = n​
m+1 = n+1 implies m = n​
?

Homework Helper

#### kenewbie

I just don't get this. This is exactly the problem I have with some of the "proofs" that I am given to construct in textbooks, they are obvious before the fact, and that is a horrible way to teach a concept.

m + k = n + k | -k
m = n

Of course that is true! there is no natural number k so that 3 + k = 4 + k!

I have a mental block or something which prohibits me from learning a method by examples where I don't actually need to use the method to get the desired result!

End-of-rant

k

#### Wretchosoft

If you're allowed to, use the fact there is an ordering on the natural numbers.

#### lanedance

Homework Helper
i know what you mean kenewbie, i'm struggling with the same issue on analysis proofs... but there in lies the twist, if you can't prove something that appears obvious, how can you prove something less obvious, that probably relies on your obvious friend?

i think its about breaking it right down to what actually is assumed eg. your axioms... and what follows as a logical consequence from those axioms alone. Otherwise seemingly simple assumptions could get swept up in your reasoning without a concrete base for them

probably wrong place to hijack somones question to discuss it, but interested on anyone else's thoughts on this

#### Chaos2009

Well, your first example uses k = 0. Your next step you said that your k was 1 greater than your previous value of k, k = k + 1. If you prove that this next step is also true and that this process of getting to the next natural number is valid, then you just proved it, I think. I haven't done much of this kind of problem.

#### HallsofIvy

Homework Helper
I asked before, what definition of addition are you using? I don't see how you can prove anything about addition of natural numbers without using the basic definition.

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