# Thread: Is this notation valid ?

1. ## Is this notation valid ?

Hi all,
let $\mathbb{W}_t$ be a subset of $\mathbb{W}$ for any $t$ (its elements are defined by $t$). If I want to say that if I performed union of every set $\mathbb{W}_t$ with $1 \leqslant t \leqslant k$, I would obtain $\mathbb{W}$, is it valid notation to write :

$\displaystyle \mathbb{W} = \bigcup_{t=1}^{k} \mathbb{W}_t$

I don't want to use the heavy $\mathbb{W} = \mathbb{W}_1 \cup \mathbb{W}_2 \cup \dots \cup \mathbb{W}_k$ notation, is the shortcut above valid and understandable ?

Thanks all

2. Originally Posted by Bacterius
Hi all,
let $\mathbb{W}_t$ be a subset of $\mathbb{W}$ for any $t$ (its elements are defined by $t$). If I want to say that if I performed union of every set $\mathbb{W}_t$ with $1 \leqslant t \leqslant k$, I would obtain $\mathbb{W}$, is it valid notation to write :

$\displaystyle \mathbb{W} = \bigcup_{t=1}^{k} \mathbb{W}_t$

I don't want to use the heavy $\mathbb{W} = \mathbb{W}_1 \cup \mathbb{W}_2 \cup \dots \cup \mathbb{W}_k$ notation, is the shortcut above valid and understandable ?

Thanks all
Yes. Not only is it valid and understandable, that is the standard notation for what you are trying to do.

3. Yes, this notation is fine. As a minor point, the previous discussion should make it clear that t is an integer, not a real.

4. Yes, it is an integer, I often forget to add obvious stuff like this although I know I shouldn't
Thanks for your replies !!

5. 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.

6. Sorry I am confused by the above post what function is 'W' being used to symbolise?

7. The function whose domain is some set of indices, of which 1 through k are members, and whose value is Wt for each t in the domain.

8. [QUOTE=Bacterius;592350]Hi all,
If I want to say that if I performed union of every set $\mathbb{W}_t$ with $1 \leqslant t \leqslant k$, I would obtain $\mathbb{W}$, is it valid notation to write :

is this what you are talking about? (sorry I am still a bit confused)

9. Thanks Moeblee for the detailed reply, I appreciate it

Hmmmm, $\mathbb{W}$ would be a non-empty set of natural integers (for instance), and $\mathbb{W}_t$ represents a subset of $\mathbb{W}$, whose elements are chosen as a function of $t \in \mathbb{N}$. A crude example would be :

Originally Posted by Example
$1 \leq t \leq 3$, and $\mathbb{W} = \{4, 11, 15, 16, 71\}$

$\mathbb{W}_t = \{4, 11, 15\}$ if $t = 1$, $\mathbb{W}_t = \{15, 71\}$ if $t = 2$ and $\mathbb{W}_t = \{4, 16\}$ if $t = 3$.
And I just wanted to know if it was correct (syntaxically speaking) to write :

$\displaystyle \bigcup_{t=1}^{3} \mathbb{W}_t = \mathbb{W}_1 \cup \mathbb{W}_2 \cup \mathbb{W}_3 = \mathbb{W}$

For any $k \in \mathbb{N}$ - for the example $k = 3$.

10. 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.

11. MoeBlee I was replying to "hmmmm" (that was awkward, no pun intended)
Thanks for clarifying though, I'll make good use of it

12. yeah i wasnt confused by the notation i was asking moeblee for clarification of the point about W being used in two ways because I didnt understand sorry i think I may have just made the thread a bit confusing but thanks anyway