Ordered Pairs
Why
We want to order two objects.
Definition
Let $a$ and $b$ denote objects. The ordered pair of $a$ and $b$ is the set $\set{\set{a}, \set{a, b}}$. The first coordinate of $\set{\set{a}, \set{a, b}}$ is the object denoted by $a$ and the second coordinate is the object denoted by $b$.
Notation
We denote the ordered pair $\set{\set{a},\set{a,b}}$ by $(a, b)$.
Equality
Our intuition of two objects in order dictates that if we have the same objects in the same order then we have the same ordered pair. Conversely, if we have two identical ordered pairs, they must consist of the same objects in the same location. In other words, two ordered pairs should be equal if and only if they consist of the same objects in the same order. Our definition agrees with this intuition. Indeed,
- The proof of this proposition will be found in future editions. ↩︎