Set numbering representation

Question

I want to represent proper element of sets.

For example, $$ A=\{1,2,3,4\} $$ $$A(2)=2$$ But I wonder that this expression is right. Because I know that the set has not order. How can I do this??

Answer 1

You (almost) can't. You are thinking of sets as subsets of some ordered set, like the integers or the reals. If $A=\{1,\text{ahdwhelrj},<\text{some picture file}>\}$, what is $A(2)?$ As there is no natural order, you can't distinguish this from $A=\{\text{ahdwhelrj},<\text{some picture file}>,1\}$ If you have a global axiom of choice, you can well-order the universe and then you can order any set, then define a unique function from ordinals into the set, but I suspect that is not in the range you are asking about.

Answer 2

If you only use this type of set, then this is impossible, because sets have no order, as you said. But you can use other objects, which are often helpful:

use tupels (or vectors, which are basically the same). also, you can instead use $A$ as a function: $A : \{1,2,3,4\} \to \mathbb R, \, A(k) = k$. then you can freely "access" the second element.

Answer 3

A thing that maps indices to objects is called a tuple or a sequence. It's richer than a set, because it is ordered, and it can include an object more than once. You typically use round parentheses to write the tuple and subscripts to write its elements: $$A=(1,2,3,4)$$ $$A_2=2$$