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?
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asked Apr 19, 2015 at 21:47
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1$\begingroup$ Explicit is better than implicit. Use $\wedge$. $\endgroup$– ncmathsadistApr 19, 2015 at 21:58
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$\begingroup$ Also, I would write $\{x: (x > 3)\wedge(x\not=10)\}$. $\endgroup$– ncmathsadistApr 19, 2015 at 21:58
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3$\begingroup$ Better yet, just use the word "and". I personally find unnecessary use of notation from mathematical logic to get old very quickly. Perhaps it's just me, but it makes me feel like I'm reading Principia Mathematica when we're just doing ordinary algebra... $\endgroup$– pjs36Apr 19, 2015 at 21:59
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$\begingroup$ I'd just write "and", as @pjs36, besides his point, I think the wedge is the most ugly logical conective :-p. $\endgroup$– YoTengoUnLCDJan 27, 2016 at 17:26
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2 Answers
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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:
Separating two or several variables that satisfy the same criterion ($p,q \in \Bbb Z$).
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.
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$\begingroup$ I saw those exact examples on Wikipedia and then came here wondering whether the two symbols are really just synonyms. $\endgroup$– MusefulSep 20, 2018 at 18:39
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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.
answered Apr 19, 2015 at 21:56