Math 2.0

Mathematical logic and foundations Proof

Proof of the commutativity of addition by induction

Builds on Addition of natural numbersSuccessor functionMathematical induction

Proof

We argue by induction, first for two lemmas and then for the statement. Let SS be the successor function. The addition satisfies a+0=aa + 0 = a and a+S(b)=S(a+b)a + S(b) = S(a + b) for all natural numbers aa and bb. Write 1=S(0)1 = S(0). Then n+1=n+S(0)=S(n+0)=S(n)n + 1 = n + S(0) = S(n + 0) = S(n) for each natural number nn.

  1. Lemma 1: 0+a=a0 + a = a for every natural number aa. We use induction on aa.
  2. For a=0a = 0, the definition gives 0+0=00 + 0 = 0.
  3. Suppose that 0+a=a0 + a = a. Then 0+S(a)=S(0+a)=S(a)0 + S(a) = S(0 + a) = S(a), by the definition and the hypothesis. This proves Lemma 1.
  4. Lemma 2: S(a)+b=S(a+b)S(a) + b = S(a + b) for all natural numbers aa and bb. Fix aa and use induction on bb.
  5. For b=0b = 0, we have S(a)+0=S(a)S(a) + 0 = S(a) and S(a+0)=S(a)S(a + 0) = S(a), so both sides are equal.
  6. Suppose that S(a)+b=S(a+b)S(a) + b = S(a + b). Then S(a)+S(b)=S(S(a)+b)=S(S(a+b))S(a) + S(b) = S(S(a) + b) = S(S(a + b)), by the definition and the hypothesis.
  7. Also S(a+S(b))=S(S(a+b))S(a + S(b)) = S(S(a + b)) by the definition. So S(a)+S(b)=S(a+S(b))S(a) + S(b) = S(a + S(b)), which proves Lemma 2.
  8. Now fix aa. We prove a+b=b+aa + b = b + a by induction on bb.
  9. For b=0b = 0, we have a+0=aa + 0 = a by the definition and 0+a=a0 + a = a by Lemma 1. So a+0=0+aa + 0 = 0 + a.
  10. Suppose that a+b=b+aa + b = b + a. Then a+S(b)=S(a+b)=S(b+a)a + S(b) = S(a + b) = S(b + a), by the definition and the hypothesis.
  11. By Lemma 2, S(b)+a=S(b+a)S(b) + a = S(b + a). So a+S(b)=S(b)+aa + S(b) = S(b) + a.
  12. By induction, a+b=b+aa + b = b + a for every natural number bb. Since aa was arbitrary, this holds for all natural numbers aa and bb.

Idea

The definition of addition only tells us how to add 00 or a successor on the right. The two lemmas show that 00 and successors behave in the same way on the left. With them, an induction on the second number moves each successor from one side to the other.

Reading path

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Definition by recursion, concept, no evidence levelDefinition by recursi…conceptMathematical induction, concept, no evidence levelMathematical inductionconceptNatural numbers, object, no evidence levelNatural numbersobjectPeano axioms, concept, no evidence levelPeano axiomsconceptSuccessor function, definition, no evidence levelSuccessor functiondefinitionAddition of natural numbers, definition, no evidence levelAddition of natural n…definitionProof of the commutativity of addition by induction, proof, no evidence levelProof of the commutat…proof
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  1. Definition by recursion concept
  2. Mathematical induction concept
  3. Natural numbers object
  4. Peano axioms concept
  5. Successor function definition
  6. Addition of natural numbers definition
  7. Proof of the commutativity of addition by induction proof

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Further reading

  1. Web https://en.wikipedia.org/wiki/Proofs_involving_the_addition_of_natural_numbers, Proof of commutativity
About this page
Tag4EQPCHKEBH
URIm2://4EQPCHKEBH
Kindproof
WritingW0 stub
MSC03F30
ProvesCommutativity of addition
UsesAddition of natural numbers, Successor function, Mathematical induction
MethodsMathematical induction
Changed4 November 2026

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