Arithmetic Addition is Commutative

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Theorem

Let $\mathbb F$ be one of the standard number sets: $\N, \Z, \Q, \R$ and $\C$.

Then:

$\forall x, y \in \mathbb F: x + y = y + x$

That is, the operation of addition on the standard number sets is commutative.


Natural Number Addition is Commutative

The operation of addition on the set of natural numbers $\N$ is commutative:

$\forall m, n \in \N: m + n = n + m$


Integer Addition is Commutative

The operation of addition on the set of integers $\Z$ is commutative:

$\forall x, y \in \Z: x + y = y + x$


Rational Addition is Commutative

The operation of addition on the set of rational numbers $\Q$ is commutative:

$\forall x, y \in \Q: x + y = y + x$


Real Addition is Commutative

The operation of addition on the set of real numbers $\R$ is commutative:

$\forall x, y \in \R: x + y = y + x$


Complex Addition is Commutative

The operation of addition on the set of complex numbers is commutative:

$\forall z, w \in \C: z + w = w + z$