let be a subset of for any (its elements are defined by ). If I want to say that if I performed union of every set with , I would obtain , is it valid notation to write :
I don't want to use the heavy notation, is the shortcut above valid and understandable ?
Rather than say "defined by t", I would say instead "t is the index".
Otherwise, from a practical, everyday working point of view, this is accepted notation, and is commonly found in mathematics, even among logicians and set theorists.
However, from a more strict, albeit pedantic, point of view, the notation is flawed: 'W' is being used in two different and incompatible ways. First it's used to symbolize a certain function, then it is used to symbolize a particular subset of the union of the range of said function.
Usually, for purposes of mathematical communication, there is no harm in such usage, but from a more strictly formal point of view, the usage is flawed in the way I mentioned.
Thanks Moeblee for the detailed reply, I appreciate it
Hmmmm, would be a non-empty set of natural integers (for instance), and represents a subset of , whose elements are chosen as a function of . A crude example would be :
And I just wanted to know if it was correct (syntaxically speaking) to write :Originally Posted by Example
For any - for the example .
Yes, that's all correct. And your use of 'W' in that way is well understood among mathematicians including set theorists. My only point though is that from an extremely technical point of view, the symbol 'W' should not be used both for the function itself and the union that you mentioned. But, again, that is only from a very technical point of view that is not of concern in ordinary, everyday mathematical communication.