# Why do some authors define the ordered pair as the set: $(a,b)=\{\{a\},\{a,b\}\}$? [duplicate]

I am using a textbook called foundations of mathematical analysis by johnsonbaugh and in it, he defines the ordered pair of elements $a$ and $b$, writen as $(a,b)$ as the set:

$(a,b)=\{\{a\},\{a,b\}\}$ where $a$ is called the first element of $(a,b)$ and $b$ is called the second element of $(a,b)$.

This definition is a bit strange, would anyone know how I can interpret it? Thank you!

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## marked as duplicate by Henning Makholm, Pedro Tamaroff♦, egreg, mrf, Sami Ben Romdhane Apr 6 '14 at 22:23

There's no need to keep alternating into and out of MathJax. I've set your notation entirely within MathJax. Take a look. – Michael Hardy Apr 6 '14 at 21:19
Perhaps it is worth mentioning that originally this definition is due to Kazimierz Kuratowski. – Tomek Kania Apr 6 '14 at 21:21
– Henning Makholm Apr 6 '14 at 21:25

Sets don't have order. For example, $\{1, 2, 3 \} = \{2, 3, 1 \}$. One possible trick to preserve order is to use the definition above. Simply say that the item that's an element of both elements of the set is the first element in the pair, and the item that appears in just one is the second element.

Edit: Some motivation for expressing ordered pairs as sets is the fact that most of mathematics is built up from set theory.

Edit 2: I figured I'd present two alternate definitions of an ordered pair that also work, but are a bit uglier to actually use. $$(a, b) := \{\{a, 0\}, \{b, 1\}\} \\ (a, b) := \{a, \{a, b\} \}$$

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