# Ordered sets - can there be two of the same element? (set theory)

In set theory, can you have an ordered set which contains the same element? For instance, if you have a cartesian product which has an ordered pair of $\langle a,a\rangle$, do you keep these as two elements in the ordered pair? Or do we write this as $\langle a\rangle$, in the same way that for basic sets we would - $\{a,a\}$ can be written as $\{a\}$?

You can definitely have an ordered pair (or ordered tuple) with the same element, and you should definitely write that element twice (or more).

Tuples are not sets in the sense that order and repetition matter. So you can't just ignore these like you do with sets.

• Shouldn't it say that repetition and order does matter, when it comes to tuples? – Stefan Mesken Feb 9 '16 at 21:30
• But... That's what I wrote. – Asaf Karagila Feb 9 '16 at 21:40
• You're right. I'm not even sure anymore, how I managed to misinterpret your line. – Stefan Mesken Feb 9 '16 at 21:45
• It's the severe case of you not coming to the conference next week. – Asaf Karagila Feb 9 '16 at 21:49

I think the fact that, in the first sentence, you refer to $(a,a)$ as an 'ordered set' is where the confusion lies. In the context of cartesian products just use 'ordered pairs', then since $(a,a)$ is not actually a set, the repeated elements rule does not apply and so you would keep it as $(a,a)$

• Why would you say that $<a,a>$ isn't a set? – Stefan Mesken Feb 9 '16 at 21:28
• Well with ordered pairs you should use $(a,a)$ as the notation – Connor Bishop Feb 9 '16 at 21:51
• Both notations are quite common, but that's beside the point. – Stefan Mesken Feb 9 '16 at 21:55