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Two disjoint sets. In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. [1] For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two ...
One set is said to intersect another set if . Sets that do not intersect are said to be disjoint . The power set of X {\displaystyle X} is the set of all subsets of X {\displaystyle X} and will be denoted by ℘ ( X ) = def { L : L ⊆ X } . {\displaystyle \wp (X)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{~L~:~L\subseteq X~\}.}
A disjoint union of an indexed family of sets (:) is a set , often denoted by , with an injection of each into , such that the images of these injections form a partition of (that is, each element of belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their union.
In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} and { 3 , 4 } {\displaystyle \{3,4\}} is { 1 , 2 , 4 ...
One family of algorithms, known as path compression, makes every node between the query node and the root point to the root. Path compression can be implemented using a simple recursion as follows: function Find(x) is if x.parent ≠ x then x.parent := Find(x.parent) return x.parent else return x end if end function
A partition of a set X is a set of non-empty subsets of X such that every element x in X is in exactly one of these subsets [2] (i.e., the subsets are nonempty mutually disjoint sets). Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold: [3]
That is, for any sets ,, and , one has = () = () Inside a universe , one may define the complement of to be the set of all elements of not in . Furthermore, the intersection of A {\displaystyle A} and B {\displaystyle B} may be written as the complement of the union of their complements, derived easily from De Morgan's laws : A ∩ B = ( A c ...
We say that two sets A and B are lattice disjoint or disjoint if a and b are disjoint for all a in A and all b in B, in which case we write . [2] If A is the singleton set { a } {\displaystyle \{a\}} then we will write a ⊥ B {\displaystyle a\perp B} in place of { a } ⊥ B {\displaystyle \{a\}\perp B} .