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So we are told to do induction on this statement:

"Let P(n) be the statement: If a + n = c + n, then a = b.", and I am told to induct on n, so I do.

Base: Let P(0) be the statement : "If a + 0 = c + 0, then a = b". which is true.

From there, you assume P(n) to be true for n (denoted n'). However, I'm stuck on how to go on from here as well as what my inductive hypothesis is.

Thanks all!

Lukas Heger
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Joesteffy
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  • Your induction hypothesis is that $a+n = b+n$ implies $a=b$ and you want to show that $a+n+1 = b+n+1$ implies $a=b$. But what are you working with? Is addition commutative? Can't you just subtract $n$? – 57Jimmy Feb 01 '18 at 17:35
  • @57Jimmy Subtracting n is really adding -n. So this would amount to arguing that adding -n to both sides preserves the equality. But "adding the same thing to both sides of equation preserves the equality" is precisely what's being asked to prove. – Acccumulation Feb 01 '18 at 17:57
  • @Acccumulation I most certainly agree, but since the OP put "algebraic number theory" as a tag, I was just wondering what he was possibly trying to do. – 57Jimmy Feb 01 '18 at 18:00
  • Your title is rather generic; induction is generally used to show that a property always follows from certain conditions, so "induction" and "if then" is redundant. – Acccumulation Feb 01 '18 at 20:33

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Any time you're trying to do a proof, you need to look at what your premises are. If you're trying to prove this in Peano arithmetic, then you have the axiom that (a = b) => (a+1=b+1). So if (a+n)=(b+n), then (a+n)+1=(b+n)+1, and by associativity a+(n+1)=b+(n+1).

Acccumulation
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