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I'm looking for a widely used notation to describe the first element of a set.


S = {5, 7,...,123} is the set, and obviously 5 is the first element. S = {345, 123,...,33} is the set, and obviously 345 is the first element. Exists a symbol to represent both?

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For sets, it doesn't make any sense to define the first element, because (for example) the sets $\{0,1\}$ and $\{1,0\}$ are the same. – Stefan Nov 26 '12 at 20:15
A set has no inherent order and therefore has no first element. In order for first element to make sense, you need to be talking about sequences or tuples of some kind. – Brian M. Scott Nov 26 '12 at 20:16
Generally sets are viewed without order. $\{1,2,3\} = \{3,2,1\}$ etc. – Tom Oldfield Nov 26 '12 at 20:16
unless, you're talking about an array or tuple in programming, mathematically a set is just a bowl of elements and it has no first or last element, the funny thing is that this bowl can still take more of the same elements it contained and remain unchanged. – user31280 Nov 26 '12 at 20:41
Perhaps you want to look at partially ordered sets. In order theory and lattice theory a common way to denote the smallest element (if it exists) is \bot. – Ittay Weiss Nov 26 '12 at 22:28
up vote 5 down vote accepted

Sets are collections of mathematical objects without importance to the order or repetition. That is $$\{0,0,0,0,0,0,1,1,1,1,2,2,2,2,1,1,1,1,0\}=\{0,1,2\}=\{2,2,1,0\}=\ldots$$

If you are interested in the order then you wish to talk about sequences rather than sets. Sequences are often denoted by $\langle a_i\mid i\in I\rangle$ where $I$ is an index set which carries (usually) some natural order, at least in the case of sequences. For example $I$ can be taken as the natural numbers or a finite subset of them. If the index set is very small we can just write the sequences as $\langle a_1,\ldots,a_n\rangle$.

So we have $\langle 21,34,42\rangle$. We can treat this as a function from $\{1,2,3\}$ into some other set, that is $h(1)=21, h(2)=34, h(3)=42$. Then we can write $h(1)$ or $h_1$ for the first element of the sequence.

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