I recently found a book left on the AMRC coffee table “Fundamental Concepts of Mathematics” by R.L. Goodstein. Inside his book Goodstein propose to use mathematical induction to show that addition is commutative.

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#### We start by defining addition

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We first need to define the addition of two number as satisfying the following property: (this is associativity with under addition). Let us refer this property as .

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#### Showing associativity for all integers

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We can first show that associativity is always true. We write the property .

The proof by induction is:

1) we know to be true for as it give

2) let us assume it exists an integers satisfying so that we can write .

It follows that:

using

using

since is true for

using

We now see that if satisfies then the next integer satisfies . Since is true for it is true for all positive integers.

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#### Showing that addition with is commutative

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From here we can show commutativity with . We refer to this property as : for all (positive) integers.

The proof by induction is the following:

1) we know that is true for , as we simply have

2) let us assume that it exists an integer for which is true. Then we can write .

Since satisfies

using

We now see that if satisfy then the next integer satisfy . Since is true for it is true for all positive integers.

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#### Showing that addition is commutative for all integers

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Next we can show the property : commutativity with any integer .

The proof by induction is:

1) we know is true for as it gives which simply is which we now know to be true.

2) let us assume that is true for some integer then we can write .

It follows that

using

since is true for

using

using

using

So that we see that if satisfy then the next integer satisfy . Therefore is true for all integers.