# Product of Cardinals is Associative

## Theorem

Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be cardinals.

Then:

$\mathbf a \paren {\mathbf {b c} } = \paren {\mathbf {a b} } \mathbf c$

where $\mathbf {a b}$ denotes the product of $\mathbf a$ and $\mathbf b$.

## Proof

Let $\mathbf a = \card A$, $\mathbf b = \card B$ and $\mathbf c = \card C$ for some sets $A$, $B$ and $C$.

By definition of product of cardinals:

$\mathbf a \paren {\mathbf {b c} }$ is the cardinal associated with $A \times \paren {B \times C}$.

Consider the mapping $f: A \times \paren {B \times C} \to \paren {A \times B} \times C$ defined as:

$\forall a \in A, b \in B, c \in C: \map f {a, \tuple {b, c} } = \tuple {\tuple {a, b}, c}$

Let $a_1, a_2 \in A, b_1, b_2 \in B, c_1, c_2 \in C$ such that:

$\map f {a_1, \tuple {b_1, c_1} } = \map f {a_2, \tuple {b_2, c_2} }$

Then:

 $\ds \map f {a_1, \tuple {b_1, c_1} }$ $=$ $\ds \map f {a_2, \tuple {b_2, c_2} }$ $\ds \leadsto \ \$ $\ds \tuple {\tuple {a_1, b_1}, c_1}$ $=$ $\ds \tuple {\tuple {a_2, b_2}, c_2}$ Definition of $f$ $\ds \leadsto \ \$ $\ds \tuple {a_1, b_1}$ $=$ $\ds \tuple {a_2, b_2}$ Equality of Ordered Tuples $\, \ds \land \,$ $\ds c_1$ $=$ $\ds c_2$ $\ds \leadsto \ \$ $\ds a_1$ $=$ $\ds a_2$ Equality of Ordered Tuples $\, \ds \land \,$ $\ds b_1$ $=$ $\ds b_2$ $\, \ds \land \,$ $\ds c_1$ $=$ $\ds c_2$ $\ds \leadsto \ \$ $\ds a_1$ $=$ $\ds a_2$ Equality of Ordered Tuples $\, \ds \land \,$ $\ds \tuple {b_1, c_1}$ $=$ $\ds \tuple {b_2, c_2}$ $\ds \leadsto \ \$ $\ds \tuple {a_1, \tuple {b_1, c_1} }$ $=$ $\ds \tuple {a_2, \tuple {b_2, c_2} }$ Equality of Ordered Tuples

Thus $f$ is an injection.

 $\ds \forall x \in \paren {A \times B} \times C: \exists a \in A, b \in B, c \in C: \,$ $\ds x$ $=$ $\ds \tuple {\tuple {a, b}, c}$ $\ds$ $=$ $\ds \map f {a, \tuple {b, c} }$

Thus $f$ is a surjection.

So $f$ is both an injection and a surjection, and so by definition a bijection.

Thus a bijection has been established between $A \times \tuple {B \times C}$ and $\tuple {A \times B} \times C$.

It follows by definition that $A \times \tuple {B \times C}$ and $\tuple {A \times B} \times C$ are equivalent.

The result follows by definition of cardinal.

$\blacksquare$