Wikipedia:Reference desk/Archives/Mathematics/2020 November 5

= November 5 =

Set constructors
Some time ago I have used an expression $$\{t\in Q: t\cup x\}$$ meaning "build a new set from elements t of the set Q by adding x to each t " (with x defined elsewhere). This has been changed into $$\{t\cup x: t\in Q\}$$ – as I understand, "take a set of sums of t and x, where t are taken from Q".

Recently I found an expression similar to mine in ordering of clauses, but using a vertical bar instead of a colon: $$ X_{\leq(x)} = \{ y \in X \mid y \leq x\},$$ which clearly means "take each element y of the set X which is less than x".

Both kind of expressions define a set – the latter by filtering X by some condition, the former by applying an expression to each element of Q and collecting results. Are these two kind of expressions defined somewhere? How exactly do one read them (translate into a spoken English)? Did I actually made a mistake in the first expression quoted above? CiaPan (talk) 10:52, 5 November 2020 (UTC)
 * The Axiom schema of specification is what allows you to create sets like this. Very roughly it says if P is some predicate, meaning a function P(x) which evaluates to True or False for a given input x, and a set A, then you can form a new set {x∈A: P(x)}. This is read "The set of x in A such that P(x)". You can use the pipe | instead of the color : with no change in meaning, it's just a matter of individual preference. There are a few shortcuts with this; for example if it's clear that the only x's that are being discussed must be in a given set then you can drop the "∈A". For example if you're talking about real numbers then you can use {x: x<1) as short hand for {x∈R: x<1). A somewhat more complicated construction, actually using the Axiom schema of replacement, uses an expression in front of the colon. In this case {e(x): x∈A} is short hand for {y: y = e(x) for some x∈A}. For any set expression you must be able to tell when something is in the set. For B={x∈A: P(x)} then x∈B exactly when x∈A and P(x). For C={e(x): x∈A}, y∈C exactly when there is an x∈A so that e(x) = y. So in your first example, y∈{t∪x: t∈Q) exactly when there is some t∈Q so that y=t∪x. In the second example, y∈X≤(x) exactly when y∈X and y≤x. Note that you must always have some containing set A for the Axiom specification; this is to avoid issues such as Russell's paradox. --RDBury (talk) 15:35, 5 November 2020 (UTC)
 * To answer user CiaPan's last question directly: Yes, the notation for the first expression was not in accordance with the usual conventions. If $$S$$ is a set, and $$P$$ is a predicate defined on $$S$$, then $$\{x\in S:P(x)\}$$ is a subset of $$S.$$ This does not pattern match against your first expression, because the expression following the colon is not truth-valued. The other kind of notation for set comprehension involves two sets $$S$$ and $$T,$$ and a function $$f:S\to T.$$ Then the expression $$\{f(x)\mid x\in S\}$$ stands for a subset of $$T.$$ This pattern is used by the expression $$\{t\cup x:t\in Q\}$$ (but using a colon). --Lambiam 16:31, 5 November 2020 (UTC)
 * For the first question, see Set-builder notation. For the second question, when lecturing, I'd enounce these expressions in English as:
 * $$\{t\cup x\mid t\in Q\}$$: the set of sets of the form $$t$$ union $$x$$ where $$t$$ is an element of $$Q$$;
 * $$\{y\in X:y\leq x\}$$: the set of all $$y$$ in big $$X$$ such that $$y$$ is at most small $$x$$.
 * Depending on the context and audience I might abbreviate this; e.g., for the last one, to: "the $$y$$ in big $$X$$ that are at most small $$x$$". Or I might use more elaborate language: "the subset of big $$X$$ consisting of all elements $$y$$ of big $$X$$ such that ...". See also Language of mathematics.  --Lambiam 16:59, 5 November 2020 (UTC)
 * As far as I'm aware, there is no distinction between the colon and the vertical line. It's just a variant notation. --Trovatore (talk) 21:24, 5 November 2020 (UTC)

Thank you all very much for your valuable input! Happy editing! CiaPan (talk) 08:59, 10 November 2020 (UTC)