Union

The union of the argument collections, sorted and de-duplicated; lists give a list, sets a set.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaUnion (set theory)↗MathWorldUnion↗WikidataQ185359↗Britannicatopic/union-set-theory↗FungrimUnion↗Wolfram LanguageUnion!↗

Union(a, b, …)the sorted, de-duplicated union of the lists, as a $List$ (of sets, as a $Set$).

Typed signatures

Type	Package	Applies
(any+) -> collection	combinatoricsreplaces compute-engine	always
(any+) -> set	compute-engine	always

Domain: Collections

Details

• Duplicates within and across all the argument collections are removed. Lists in give a list in Wolfram's canonical order (shorter sublists first); sets in give a set. With no arguments the result is $EmptySet$ (Wolfram's Union[] is the empty list).
• Commutative and idempotent: $A \cup A = A$, and argument order doesn't affect the result.
• Inclusion-exclusion: $|A \cup B| = |A| + |B| - |A \cap B|$. See Intersection and Length.
• Sorted, matching Wolfram's Union — compute-engine's own de-duplication preserves first-seen order instead.

Examples

Basic #

live ↗

In$\bigl\lbrack1, 2, 3\bigr\rbrack∪\operatorname{List}(3, 4)$Out$\bigl\lbrack1, 2, 3, 4\bigr\rbrack$

live ↗

In$\operatorname{List}(1, 2)∪\operatorname{List}(2, 3)∪\operatorname{List}(3, 4)$Out$\bigl\lbrack1, 2, 3, 4\bigr\rbrack$

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In$\bigl\lbrack a, b, c\bigr\rbrack∪\bigl\lbrack b, c, d\bigr\rbrack$Out$\bigl\lbrack a, b, c, d\bigr\rbrack$

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In$\bigl\lbrack a, b, a, c\bigr\rbrack∪\bigl\lbrack d, a, k, b\bigr\rbrack∪\operatorname{List}(c, a)$Out$\bigl\lbrack a, b, c, d, k\bigr\rbrack$

Scope #

A single list is de-duplicated

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In$\mathrm{Union}(\bigl\lbrack1, 2, 1, 3, 6, 2, 2\bigr\rbrack)$Out$\bigl\lbrack1, 2, 3, 6\bigr\rbrack$

Symbolic elements, merged across three lists

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In$\bigl\lbrack a, b, a, c\bigr\rbrack∪\bigl\lbrack d, a, f, b\bigr\rbrack∪\operatorname{List}(c, a)$Out$\bigl\lbrack a, b, c, d, f\bigr\rbrack$

Elements can themselves be lists; $\{1, 2\}$ and $\{2, 1\}$ stay distinct

live ↗

In$\operatorname{List}(\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack1, 2, 3\bigr\rbrack)∪\operatorname{List}(\bigl\lbrack2, 1\bigr\rbrack, \bigl\lbrack1, 2\bigr\rbrack)∪\operatorname{List}(\bigl\lbrack3, 2, 1\bigr\rbrack, \bigl\lbrack1, 2, 3\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack2, 1\bigr\rbrack, \bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack3, 2, 1\bigr\rbrack\bigr\rbrack$

With no arguments, the union is empty

live ↗

In$$Out$∅$

Sets in, a set out

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In$\lbrace3, 1\rbrace∪\lbrace2, 3\rbrace$Out$\lbrace1, 2, 3\rbrace$

Lists in, a sorted de-duplicated list out, as in Wolfram

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In$\bigl\lbrack3, 1, 3\bigr\rbrack∪\bigl\lbrack2\bigr\rbrack$Out$\bigl\lbrack1, 2, 3\bigr\rbrack$

Applications #

Combining the divisors of 10 and 15 into one set

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In$\mathrm{Divisors}(10)∪\mathrm{Divisors}(15)$Out$\bigl\lbrack1, 2, 3, 5, 10, 15\bigr\rbrack$

Combining the divisors of 10, 12 and 20 into one sorted set

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In$\mathrm{Divisors}(10)∪\mathrm{Divisors}(12)∪\mathrm{Divisors}(20)$Out$\bigl\lbrack1, 2, 3, 4, 5, 6, 10, 12, 20\bigr\rbrack$

Counting the distinct values in a list. See Length

live ↗

In$\mathrm{Length}(\mathrm{Union}(\bigl\lbrack1, 2, 1, 3, 6, 2, 2\bigr\rbrack))$Out$4$

Seven of the eight possible 3-bit windows occur in the binary digits of $10!$; $001$ never does

live ↗

In$\mathrm{Union}(\mathrm{Partition}(\mathrm{IntegerDigits}(10!, 2), 3, 1))$Out$\bigl\lbrack\bigl\lbrack0, 0, 0\bigr\rbrack, \bigl\lbrack0, 1, 0\bigr\rbrack, \bigl\lbrack0, 1, 1\bigr\rbrack, \bigl\lbrack1, 0, 0\bigr\rbrack, \bigl\lbrack1, 0, 1\bigr\rbrack, \bigl\lbrack1, 1, 0\bigr\rbrack, \bigl\lbrack1, 1, 1\bigr\rbrack\bigr\rbrack$

Powers of $5$ mod $11$ cycle through just five residues, the quadratic residues

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In$\mathrm{Union}(\mathrm{NestList}(\operatorname{\_1}↦5\operatorname{\_1}\bmod11, 1, 100))$Out$\bigl\lbrack1, 3, 4, 5, 9\bigr\rbrack$

Properties #

$Union$ is commutative: argument order doesn't matter

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In$\bigl\lbrack1, 2, 3\bigr\rbrack∪\bigl\lbrack2, 3, 4\bigr\rbrack=\bigl\lbrack2, 3, 4\bigr\rbrack∪\bigl\lbrack1, 2, 3\bigr\rbrack$Out$\mathrm{True}$

Inclusion-exclusion: $|A \cup B| = |A| + |B| - |A \cap B|$. See Length and Intersection

live ↗

In$\mathrm{Length}(\bigl\lbrack1, 2, 3\bigr\rbrack∪\bigl\lbrack2, 3, 4\bigr\rbrack)=(\mathrm{Length}(\bigl\lbrack1, 2, 3\bigr\rbrack)+\mathrm{Length}(\bigl\lbrack2, 3, 4\bigr\rbrack))-\mathrm{Length}(\bigl\lbrack1, 2, 3\bigr\rbrack∩\bigl\lbrack2, 3, 4\bigr\rbrack)$Out$\mathrm{True}$

The empty set is the identity for $Union$: $A \cup \varnothing = A$

live ↗

In$\lbrace1, 2\rbrace∪∅$Out$\lbrace1, 2\rbrace$

About

Examples	18
Checked against	Wolfram Language 16 agree1 differ
Crosswalk	WikipediaMathWorldWikidataBritannicaFungrimWolfram Language

See also: Intersection, SetMinus