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I have an idea about how to do this. I've seen this before so this is what I think. Is this how you refer to a single element in a set?

$$a = \{5, 7, 3, 4\};a_2 = 7$$

Is this correct?

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I've never seen that notation, and it's a little strange since the sets $a=\{5,7,3,4\}=\{3,4,5,7\}$ are equal. So what is $a_2$? Is it $7$ or $4$, or something else? I would just say "an element $7\in a$" to refer to $7$. – Vika Feb 11 '12 at 2:47
No, because sets don't have order and repeats don't matter; as sets, $\{5,7,3,4\}=\{3,5,4,7\}=\{3,3,3,3,5,4,7\}$, which makes the notation ill-defined. Now, if $a$ were an ordered $4$-tuple, $a=(5,7,3,4)$, then it is common to refer to the $i$th entry of the tuple as $a_i$. – Arturo Magidin Feb 11 '12 at 2:48
@ArturoMagidin Okay, I get it now. – David Feb 11 '12 at 2:49
If $a=[5,7,3,4]$ was a list, then $a_2=7$ is good notation. However, as Vika points out, sets are unordered so referring to the "second element in the set" makes no sense. – Bill Cook Feb 11 '12 at 2:50
up vote 2 down vote accepted

You can just refer to it as 7. To say 7 is an element of the set, we write $7 \in \{5,7,3,4 \}$. Usually the order you write elements of sets in doesn't matter.

You can have very big (uncountable) sets where it is not easy to assign a number to each element of the set, for example the set of real numbers (i.e. decimal numbers) $\mathbb{R}$. So there is no (immediate) sensible idea of a "2nd element" of this set, but there is for your example.

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I should add for posterity that if you know a bit about ordinals and cardinals, it is possible to enumerate sets like $\mathbb{R}$ in such a way that every element can be put into a linear order (i.e. given any number there is a 'next' number, with no other number appearing between those two). – Paul Slevin Jun 2 '12 at 19:13

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