$\wedge$ in set builder notation

Question

Wikipedia says to use $\wedge$ in set-builder notation like $\{x \,:\, x > 3 \wedge x \neq 10\}$. However, I prefer to merely seperate predicates by a comma. Which notation is more common?

Answer 1

It may not be clear whether a comma signifies and or or. The reason it is complicated is because of the comma's use and meanings in different places: $x = 2,3$ means or, but $x\in \mathbb{R}, x>3$ may be and. So in a more ambiguous case or when your audience is unfamiliar with your notation, you should explicitly state (at least at first) what you mean, or use unambiguous notation.

Answer 2

Actually, the Wikipedia article I found uses both:

$$ \{ x \mid x \in \Bbb R \land x > 0 \} \quad\text{and}\quad \{ p/q \mid p,q \in \Bbb Z, q \neq 0 \} $$

I would suggest the former, because of the (possible) confusion that can arise in the latter example, using the comma to signify two things:

1. Separating two or several variables that satisfy the same criterion ($p,q \in \Bbb Z$).

2. Separating two clauses.

Hence, writing

$$\{ p/q \mid p,q \in \Bbb Z \land q \neq 0 \} $$

is logically consistent with what all the predicates specify together.