Equality of Cartesian Products

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Theorem

Let $A, B, C, D$ be nonempty sets.


Then:

$A \times B = C \times D \iff A = C \land B = D$

where $\times$ denotes cartesian product.


Proof

If $A = C$ and $B = D$, it is immediate that $A \times B = C \times D$.


Now suppose that $A \times B = C \times D$.

By definition of Cartesian product:

\(\displaystyle x \in A, y \in B\) \(\iff\) \(\displaystyle \tuple {x, y} \in A \times B\)
\(\displaystyle \) \(\iff\) \(\displaystyle \tuple {x, y} \in C \times D\) Definition of Set Equality
\(\displaystyle \) \(\iff\) \(\displaystyle x \in C, y \in D\)

It follows that $x \in A$ if and only if $x \in C$, and so $A = C$.

Similarly, $y \in B$ if and only if $y \in D$, hence $B = D$.


The result follows.

$\blacksquare$


Also see