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:
| \(\ds x \in A, y \in B\) | \(\leadstoandfrom\) | \(\ds \tuple {x, y} \in A \times B\) | ||||||||||||
| \(\ds \) | \(\leadstoandfrom\) | \(\ds \tuple {x, y} \in C \times D\) | Definition of Set Equality | |||||||||||
| \(\ds \) | \(\leadstoandfrom\) | \(\ds x \in C, y \in D\) |
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.
$\blacksquare$