Just for context consider something like a set $\lbrace X_i \rbrace_{i \in I}$ where $I$ is also a set. I must say I'm more familiar with the sequencial notation $(X_i)_{i \in I}$ but I guess this makes it less of a set-related question. Should we need to refer to the set whose elements are the elements of the elements of the set $\lbrace X_i \rbrace_{i \in I}$ we would use the notation $\cup_{i\in I} X_i$. However, if the set $\lbrace X_i \rbrace_{i \in I}$ was given a name, like $S$, then we would write $\cup_{s\in S} s$. But then why not write $\cup_S$ ? It seems totally unambiguous to me. I'm inclined to think that no extra information is given by that $s$ in the indices since it is $s$ itself that plays the role of the elements to be united, and not a function of $s$. Is this notation ever used ?
Asked
Active
Viewed 58 times
1
-
5For a set $S$ you may write $\bigcup S:={x:\exists A\in S$ such that $x\in A}$. For instance, if $S={A,B}$, then $\bigcup S=A\cup B$. – Renan Mezabarba Mar 22 '17 at 16:50
-
I've seen this before but I need to ask. For me, $A \cup B = \cup_{X \in \lbrace A,B \rbrace } X$. You wrote it $A \cup B = \cup \lbrace A,B \rbrace$ and this is another issue for me. – James Well Mar 22 '17 at 16:58
-
It is just notation. $\bigcup_{X\in{A,B}}X$ is also okay. – Renan Mezabarba Mar 22 '17 at 17:01
-
1In both notations, the cup's argument isn't of the same type, $X$ is a subset of the resulting set, whereas $\lbrace A,B \rbrace$ isn't. Is that not a problem ? I mean is it always clear what is meant ? Adding $S \in I$ to $\cup S$ actually changes $S$ into a subset of the whole union. – James Well Mar 22 '17 at 17:15
-
2No, it's not an issue, because in one you have a subscript and the other you have not. So there are no clashes. – martin.koeberl Mar 22 '17 at 20:19