Search results
Results From The WOW.Com Content Network
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~\}.}
function Find(x) is if x.parent ≠ x then x.parent := Find(x.parent) return x.parent else return x end if end function This implementation makes two passes, one up the tree and one back down. It requires enough scratch memory to store the path from the query node to the root (in the above pseudocode, the path is implicitly represented using ...
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]
Applying the axiom of regularity to S, let B be an element of S which is disjoint from S. By the definition of S, B must be f(k) for some natural number k. However, we are given that f(k) contains f(k+1) which is also an element of S. So f(k+1) is in the intersection of f(k) and S. This contradicts the fact that they are disjoint sets.
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} .
Luckily, any such simple set can be rewritten as a union of another finite family of rectangles, rectangles which this time are mutually disjoint, and then one defines the Jordan measure () as the sum of measures of the disjoint rectangles. One can show that this definition of the Jordan measure of is independent of the representation of as a ...
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.