1. ## Ordered Pair

A definition in my Abstract Algebra class on an ordered pair is as follows:

Let A and B be sets. Given a is an element of A and b is an element of B, the ordered pair (a,b) is the set {{a},{a,b}}.

Why isn't it {{a}, {a,b}, {b}} ? Or just {a,b}?

I guess I don't understand why {a} is included.

Any help would be greatly appreciated.

2. Originally Posted by kid funky fried
A definition in my Abstract Algebra class on an ordered pair is as follows:

Let A and B be sets. Given a is an element of A and b is an element of B, the ordered pair (a,b) is the set {{a},{a,b}}.

Why isn't it {{a}, {a,b}, {b}} ? Or just {a,b}?

I guess I don't understand why {a} is included.

Any help would be greatly appreciated.
The first reason why do we do it at all is that we want the notion of ordered pair to be defined as a set. We want to encode somehow the property of the ordered pair $\displaystyle (a,b)$ that $\displaystyle a$ comes first and $\displaystyle b$ comes second - this is how we understand what "$\displaystyle (a,b)$ is an ordered pair" means. And this is what $\displaystyle \{a,b\}$ and $\displaystyle \{\{a\}, \{a,b\}, \{b\}\}$ fail to provide ($\displaystyle \{a,b\}$ doesn't tell us that $\displaystyle a$ comes first, it is just a set containing elements $\displaystyle a$ and $\displaystyle b$, $\displaystyle \{a,b\}=\{b,a\}$)
The definition $\displaystyle \{\{a\}, \{a,b\}\}$ of the ordered pair $\displaystyle (a,b)$ is cleverly chosen. If we want to say "$\displaystyle x$ is on the first position in $\displaystyle (a,b)$" in set-theoretical language, we just say "$\displaystyle x$ is an element of all elements of $\displaystyle \{\{a\}, \{a,b\}\}$". If we want to say "$\displaystyle x$ is on the second position in $\displaystyle (a,b)$" , we say "there exists an element of $\displaystyle \{\{a\}, \{a,b\}\}$ that contains $\displaystyle x$ AND it is not true that there are two distinct elements of $\displaystyle \{\{a\}, \{a,b\}\}$ both containing $\displaystyle x$".
Moreover, the characteristic property of oredered pairs, "$\displaystyle (a,b)=(c,d)$ if and only if $\displaystyle a=c$ and $\displaystyle b=d$" remains true if we use this definition: "$\displaystyle \{\{a\}, \{a,b\}\}=\{\{c\}, \{c,d\}\}$ if and only if $\displaystyle a=c$ and $\displaystyle b=d$" (try to verify it).
This definition comes from Kuratowski and is not the only possible one, but people like it for its properties and have chosen it.

3. ## Thanks

Thank you. I appreciate you taking the time to give such a thoughtful response.