# Ordered Pair

• Sep 8th 2009, 06:15 PM
kid funky fried
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.
• Sep 9th 2009, 02:06 AM
Taluivren
Quote:

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.
• Sep 9th 2009, 02:17 PM
kid funky fried
Thanks
Thank you. I appreciate you taking the time to give such a thoughtful response.