I'm looking for a widely used notation to describe the first element of a set.
E.g.:
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?
Best Answer
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.