SetMinus

The elements of set a that are not also in set b.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaSet difference↗MathWorldSetDifference↗WikidataQ845126↗FungrimSetMinus↗Wolfram LanguageComplement!↗

SetMinus(a, b)the elements of a that are not also in b.

Typed signatures

Type	Package	Applies
(set<any>, value*) -> set	compute-engine	always

Domain: Collections

Details

• Not commutative: $A \setminus B$ generally differs from $B \setminus A$.
• Removing elements that aren't present in a leaves it unchanged.
• Identity: $A \setminus (A \setminus B) = A \cap B$. See Intersection.
• Subtracting a superset of a collapses the result to $EmptySet$.
• The result preserves encounter order rather than sorting into a canonical order.

Examples

Basic #

live ↗

In$\lbrace1, 2, 3\rbrace∖\lbrace2\rbrace$Out$\lbrace1, 3\rbrace$

live ↗

In$\lbrace1, 2, 3, 4\rbrace∖\lbrace2, 4\rbrace$Out$\lbrace1, 3\rbrace$

Removing elements that aren't present leaves the set unchanged

live ↗

In$\lbrace1, 2, 3\rbrace∖\lbrace9\rbrace$Out$\lbrace1, 2, 3\rbrace$

Several sets can be removed at once

live ↗

In$\lbrace a, b, c, d, f\rbrace∖\lbrace a, c\rbrace∖\lbrace d\rbrace$Out$\lbrace b, f\rbrace$

live ↗

In$\lbrace a, b, c, d, k\rbrace∖\lbrace a, c\rbrace∖\lbrace d\rbrace$Out$\lbrace b, k\rbrace$

Scope #

Lists as well as sets, as Wolfram's $Complement$ takes

live ↗

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

A Range as the universe to remove from

live ↗

In$(1..10)∖\lbrace2, 3\rbrace$Out$\lbrace1, 4, 5, 6, 7, 8, 9, 10\rbrace$

Properties #

$A \setminus (A \setminus B) = A \cap B$. See Intersection

live ↗

In$\lbrace1, 2, 3, 4\rbrace∖(\lbrace1, 2, 3, 4\rbrace∖\lbrace2, 4\rbrace)=\lbrace1, 2, 3, 4\rbrace∩\lbrace2, 4\rbrace$Out$\mathrm{True}$

Removing the empty set changes nothing: $A \setminus \varnothing = A$

live ↗

In$\lbrace1, 2, 3\rbrace∖∅$Out$\lbrace1, 2, 3\rbrace$

$A \setminus A = \varnothing$

live ↗

In$\lbrace1, 2, 3\rbrace∖\lbrace1, 2, 3\rbrace$Out$∅$

live ↗

In$\lbrace b, k, d, a, c\rbrace∖\lbrace b, c\rbrace$Out$\lbrace a, d, k\rbrace$

Possible issues #

$SetMinus$ is not commutative: $B \setminus A$ can differ wildly from $A \setminus B$

live ↗

In$\lbrace2, 4\rbrace∖\lbrace1, 2, 3, 4\rbrace$Out$∅$

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

live ↗

In$\lbrace5, 4, 3, 2, 1\rbrace∖\lbrace2\rbrace$Out$\lbrace1, 3, 4, 5\rbrace$

About

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

See also: Union, Intersection