Intersection

The elements common to all the argument collections.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaIntersection (set theory)↗MathWorldIntersection↗WikidataQ185837↗FungrimIntersection↗Wolfram LanguageIntersection!↗

Intersection(a, b, …)the elements common to every collection, as a $Set$.

Typed signatures

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

Domain: Collections

Details

• Idempotent: $A \cap A = A$.
• Disjoint collections intersect to the symbol $EmptySet$, not an empty $Set$.
• With three or more collections, only elements present in all of them survive.
• Same counting identity as Union, rearranged: $|A| + |B| = |A \cup B| + |A \cap B|$.

Examples

Basic #

live ↗

In$\bigl\lbrack1, 2, 3\bigr\rbrack∩\bigl\lbrack2, 3, 4\bigr\rbrack$Out$\lbrace2, 3\rbrace$

live ↗

In$\bigl\lbrack1, 2, 3, 4\bigr\rbrack∩\bigl\lbrack2, 3, 4, 5\bigr\rbrack$Out$\lbrace2, 3, 4\rbrace$

With three or more collections, only elements common to all of them survive

live ↗

In$\bigl\lbrack1, 2, 3\bigr\rbrack∩\bigl\lbrack2, 3, 4\bigr\rbrack∩\bigl\lbrack2, 3, 5\bigr\rbrack$Out$\lbrace2, 3\rbrace$

Duplicates are dropped as well

live ↗

In$\bigl\lbrack1, 1, 2, 3\bigr\rbrack∩\bigl\lbrack3, 1, 4\bigr\rbrack∩\bigl\lbrack4, 1, 3, 3\bigr\rbrack$Out$\lbrace1, 3\rbrace$

Scope #

Symbolic elements

live ↗

In$\bigl\lbrack a, b, c\bigr\rbrack∩\bigl\lbrack b, c, d\bigr\rbrack$Out$\lbrace b, c\rbrace$

A single collection comes back as a set

live ↗

In$\mathrm{Intersection}(\bigl\lbrack1, 2, 3\bigr\rbrack)$Out$\lbrace1, 2, 3\rbrace$

Applications #

The divisors common to 45 and 78

live ↗

In$\mathrm{Divisors}(45)∩\mathrm{Divisors}(78)$Out$\lbrace1, 3\rbrace$

Properties #

$Intersection$ is idempotent: $A \cap A = A$

live ↗

In$\lbrace1, 2, 3\rbrace∩\lbrace1, 2, 3\rbrace=\lbrace1, 2, 3\rbrace$Out$\mathrm{True}$

The same inclusion-exclusion count, rearranged: $|A| + |B| = |A \cup B| + |A \cap B|$. See Union

live ↗

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

Intersecting with the empty set gives the empty set: $A \cap \varnothing = \varnothing$

live ↗

In$\bigl\lbrack1, 2, 3\bigr\rbrack∩∅$Out$∅$

live ↗

In$\bigl\lbrack a, b, c\bigr\rbrack∩\bigl\lbrack d, e, f\bigr\rbrack$Out$∅$

Possible issues #

Disjoint collections intersect to the special symbol $EmptySet$, not an empty $Set$

live ↗

In$\operatorname{List}(1, 2)∩\operatorname{List}(3, 4)$Out$∅$

The result is a set, so its elements are in canonical order

live ↗

In$\bigl\lbrack c, b, a\bigr\rbrack∩\operatorname{List}(a, b)$Out$\lbrace a, b\rbrace$

About

Examples	13
Checked against	Wolfram Language 10 agree2 differ
Crosswalk	WikipediaMathWorldWikidataFungrimWolfram Language

See also: Union, SetMinus