Natural Number Multiplication is Commutative/Proof 3
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Theorem
The operation of multiplication on the set of natural numbers $\N$ is commutative:
- $\forall x, y \in \N: x \times y = y \times x$
Proof
Using the following axioms:
\((\text A)\) | $:$ | \(\ds \exists_1 1 \in \N_{> 0}:\) | \(\ds a \times 1 = a = 1 \times a \) | ||||||
\((\text B)\) | $:$ | \(\ds \forall a, b \in \N_{> 0}:\) | \(\ds a \times \paren {b + 1} = \paren {a \times b} + a \) | ||||||
\((\text C)\) | $:$ | \(\ds \forall a, b \in \N_{> 0}:\) | \(\ds a + \paren {b + 1} = \paren {a + b} + 1 \) | ||||||
\((\text D)\) | $:$ | \(\ds \forall a \in \N_{> 0}, a \ne 1:\) | \(\ds \exists_1 b \in \N_{> 0}: a = b + 1 \) | ||||||
\((\text E)\) | $:$ | \(\ds \forall a, b \in \N_{> 0}:\) | \(\ds \)Exactly one of these three holds:\( \) | ||||||
\(\ds a = b \lor \paren {\exists x \in \N_{> 0}: a + x = b} \lor \paren {\exists y \in \N_{> 0}: a = b + y} \) | |||||||||
\((\text F)\) | $:$ | \(\ds \forall A \subseteq \N_{> 0}:\) | \(\ds \paren {1 \in A \land \paren {z \in A \implies z + 1 \in A} } \implies A = \N_{> 0} \) |
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
- $\forall a \in \N_{> 0}: a \times n = n \times a$
Basis for the Induction
$\map P 1$ is the case:
\(\ds a \times 1\) | \(=\) | \(\ds a\) | Axiom $\text A$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 \times a\) | Axiom $\text A$ |
and so $\map P 1$ holds.
This is our basis for the induction.
Induction Hypothesis
Now we need to show that, if $\map P k$ is true, where $k \ge 0$, then it logically follows that $\map P {k + 1}$ is true.
So this is our induction hypothesis:
- $\forall a \in \N: a \times k = k \times a$
Then we need to show:
- $\forall a \in \N: a \times \paren {k + 1} = \paren {k + 1} \times a$
Induction Step
This is our induction step:
\(\ds a \times \paren {k + 1}\) | \(=\) | \(\ds \paren {a \times k} + \paren {a \times 1}\) | Left Distributive Law for Natural Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {k \times a} + \paren {a \times 1}\) | Induction hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {k \times a} + \paren {1 \times a}\) | Basis for the Induction | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {k + 1} \times a\) | Right Distributive Law for Natural Numbers |
The result follows by the Principle of Mathematical Induction.
$\blacksquare$
Sources
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 2.1$: Theorem $2.6$