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A nested set collection or nested set family is a collection of sets that consists of chains of subsets forming a hierarchical structure, like Russian dolls. It is used as reference concept in scientific hierarchy definitions, and many technical approaches, like the tree in computational data structures or nested set model of relational databases .
One may define the operations of the algebra of sets: union(S,T): returns the union of sets S and T. intersection(S,T): returns the intersection of sets S and T. difference(S,T): returns the difference of sets S and T. subset(S,T): a predicate that tests whether the set S is a subset of set T.
A universe set is an absorbing element of binary union . The empty set ∅ {\displaystyle \varnothing } is an absorbing element of binary intersection ∩ {\displaystyle \cap } and binary Cartesian product × , {\displaystyle \times ,} and it is also a left absorbing element of set subtraction ∖ : {\displaystyle \,\setminus :}
Euler diagram illustrating that the set of "animals with four legs" is a subset of "animals", but the set of "minerals" is a disjoint set (it has no members in common) with "animals" Euler diagram showing the relationships between different Solar System objects
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
The element w is not in the set {x, y, z} , because it hashes to one bit-array position containing 0. For this figure, m = 18 and k = 3. An empty Bloom filter is a bit array of m bits, all set to 0. It is equipped with k different hash functions, which map set elements to one of the m possible array positions.
For example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set ...
For a ≤ b, the closed interval [a, b] is the set of elements x satisfying a ≤ x ≤ b (that is, a ≤ x and x ≤ b). It contains at least the elements a and b. Using the corresponding strict relation "<", the open interval (a, b) is the set of elements x satisfying a < x < b (i.e. a < x and x < b). An open interval may be empty even if a < b.