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

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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#The grammar 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)

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