Zero is Zero Element for Natural Number Multiplication

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Theorem

Let $\N$ be the natural numbers.


Then $0$ is a zero element for multiplication:

$\forall n \in \N: 0 \times n = 0 = n \times 0$


Proof

Proof by induction.

For all $n \in \N$, let $P \left({n}\right)$ be the proposition:

$0 \times n = 0 = n \times 0$


Basis for the Induction

By definition, we have:

$0 \times 0 = 0 = 0 \times 0$

Thus $P \left({0}\right)$ is seen to be true.


This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $P \left({k}\right)$ is true, then it logically follows that $P \left({k + 1}\right)$ is true.


So this is our induction hypothesis $P \left({k}\right)$:

$0 \times k = 0 = k \times 0$


Then we need to show that $P \left({k + 1}\right)$ follows directly from $P \left({k}\right)$:

$0 \times \left({k + 1}\right) = 0 = \left({k + 1}\right) \times 0$


Induction Step

This is our induction step:


\(\displaystyle 0 \times \left({k + 1}\right)\) \(=\) \(\displaystyle \left({0 \times k}\right) + 0\) Definition of Multiplication
\(\displaystyle \) \(=\) \(\displaystyle 0 + 0\) Induction Hypothesis
\(\displaystyle \) \(=\) \(\displaystyle 0\) Definition of Addition

By definition:

$\left({k + 1}\right) \times 0 = 0$

So $P \left({k}\right) \implies P \left({k + 1}\right)$, and the result follows by the Principle of Mathematical Induction.


Therefore:

$\forall n \in \N: 0 \times n = 0 = n \times 0$

$\blacksquare$