An integer range is listed, and a family of sets has a union and an intersection

[Rafael-SOWNet]

Item 4 of the docket on #1409, second half: the constructions the chapter's exercises are written over — integer ranges, indexed unions and intersections, and the complement relative to a universe. Its rows are the tests.

ZZ /\ [1; 10] lists its members. An integer set cut by a numeric interval with at most 4096 members in it is the finite set of them (SetOperators.IntersectSpecialSetAndInterval, through the residue-class listing of #1424 with modulus 1): the book's [n] = {1, …, n} is ZZ+ /\ [1; n], [0; 2.5] /\ ZZ* is {0, 1, 2}, ZZ /\ (1; 2) is {}. The reals and the complex numbers cut by an interval are the interval; the rationals stay as written.

union(A_k, k in I) and intersection(A_k, k in I) are Entity.Set.IndexedUnionf and IndexedIntersectionf, binders like sum (name bound throughout, capture-avoiding substitution as a quantifier's), under an abstract IndexedSetOperation:

a listed index set	folded to the binary operators: union({k, 2 k}, k in ZZ+ /\ [1; 10]) is {1, …, 10, 12, 14, 16, 18, 20} (Ex 3.6.1); A_k = {k-2, …, k+2} over {1, 2, 3} unites to {-1, …, 5} and meets in {1, 2, 3} (Ex 3.6.6)
any other	an object whose membership goes through the quantifiers (#1421): x in union(A_n, n in I) is exists n in I : x in A_n. So the book's limit cases read at their members — 0 in intersection([0; 1/n), n in ZZ+) is True and 0.1 is not (§3.9.5 Try 8); 1 is in every [n] and 2 is not (Problem 3.11.20)

The families themselves stay written over an infinite index set (⋂ [0, 1/n) = {0} as a set equality needs a limit argument, not this PR's). complement(A, U) is U \ A — a complement is relative to a universe, there being no set of everything (Ex 3.5.11). Note that i is the imaginary unit, so an index is k or n; the parser says so when asked for i.

MathS.Sets.IndexedUnion/IndexedIntersection/Complement; union, intersection, complement are keywords now. Syntax.md, BREAKING-CHANGES.md (rows measured on a v2.5.0 build), PublicApi.txt regenerated, parser regenerated and post-processed. IndexedSetOperationTest, 25 rows. Full suite 12160 passed, 0 failed; benchmark gate PASSED (SimplifyHard 173,260,088 B, the same bytes as master after #1432).

What is left of item 4 is tuples and Cartesian products, which is #330's question (a pair as a value, a set of pairs) and waits for the v3 design (#1019); the docket moves to item 5, relations.

Part of #1409.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

[Happypig375]

Do e.g. max / min also want its indexed variant separate from a function of two expressions? If breaking, note for v3

[Rafael-SOWNet]

Today min and max overload on the shape of the second argument: min(a, b) is the smaller of two, min(f(t), t in S) the minimum over a set (Minimumf), decided by the second argument being an in statement with a name on its left — as argmax(f, t in S) is, and as union(A_k, k in I) is here. So the indexed variant is not separate for min/max, and it is a separate name for the set operations only because unite and intersect are infix keywords, and a unite( token would break A unite (B \/ C). Not breaking now: min(p, q in S) as "the smaller of two truth values" is nothing anyone writes, so the overload has no real ambiguity.

The v3 point is consistency of the indexed spelling across all of them — sum and product take (body, var, from, to), min/max/argmax/union/intersection take (body, var in S), and the prefix binders (forall, ∀) take var in S : body — one convention for every operation over an indexed family, with ⋃/⋂/∑ as the Unicode input of the same. Noted on #1019.