# Set builder notation: Are variables that aren't specified after the “such that” colon considered to be parameters?

Admission: I have no idea what I'm doing

I want to describe the following set (the set containing all probability distributions) in a more concise way

$$D = \{ \{ ( t_1, P(t_1 | s)), ( t_2, P(t_2 | s)), ... \text{ for all } t_i \in T \} \text{ for all } s \in S \}$$

Is there a nice way to describe this set either by set builder notation or functions? Malice Vidrine suggests

$$D = \{ \{ ( t, P(t | s)) : t \in T \} : s \in S \}$$

Are variables that aren't specified after the "such that" colon considered to be parameters?

• Why not $\{\{(t,P(t|s)):t\in T\}:s\in S\}$? – Malice Vidrine Nov 6 '19 at 4:51
• dailymail.co.uk is correct. Your suggestion is not well formed. – William Elliot Nov 6 '19 at 8:54
• That's a funny copy-paste error lol. I lack the knowledge of terminology I need to say what I'm trying to say (which is probably why I couldn't get Google to return an answer to me either). I'm sorry, I'm just a bit lost. A textbook reference would be helpful. Edit: Oh, I thought you said "question is not well formed" never mind. – bEPIK Nov 6 '19 at 10:00

## 1 Answer

There are two issues with the expression defining $$D$$.

The first one is the notation $$t_i$$, which suggests that you are working with a family, something like $$(t_i)_{1 \leqslant i \leqslant n}$$ or $$(t_i)_{i \in I}$$. If you are working with a set $$T$$, the indices $$i$$ are useless. This is why Malice Vidrine (whom you unfortunately quote incorrectly) suggested to write $$\{\{(t, P(t|s)):t \in T\}:s \in S\}$$.

The second issue is the double level in the set builder notation. Although this is mathematically correct, it would be more convenient to write

For each $$s \in S$$, let $$D_s = \left\{\left(t, P(t|s)\right):t \in T\right\}$$ --- or, if you are working with a family $$(t_i)_{i \in I}$$, $$D_s = \left\{\left(t_i, P(t_i|s)\right):i \in I\right\}$$. Then you can simply set $$D = \{D_s : s \in S\}$$.

• Cool, thanks for pointing that out – bEPIK Nov 6 '19 at 11:41