Elements of Ordered Pair do not Commute

Theorem

Let $\set {a, b}$ be a doubleton, so that $a$ and $b$ are distinct objects.

Let $\tuple {a, b}$ denote the ordered pair such that the first coordinate is $a$ and the second coordinate is $b$.

Then:

$\tuple {a, b} \ne \tuple {b, a}$

Proof

By the Kuratowski formalization of $\tuple {a, b}$:

$\tuple {a, b} = \set {\set a, \set {a, b} }$

and by Equality of Ordered Pairs:

$\tuple {a, b} = \tuple {b, a} \iff a = b$

But $a \ne b$ and so:

$\tuple {a, b} \ne \tuple {b, a}$

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 3$. Ordered pairs; cartesian product sets: Theorem $3.1$: Corollary