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In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions ...
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
Addition and multiplication are prototypical examples of operations that combine two elements of a set to produce a third element of the same set. These operations obey several algebraic laws. For example, a + (b + c) = (a + b) + c and a(bc) = (ab)c are associative laws, and a + b = b + a and ab = ba are commutative laws. Many systems studied ...
Operations can involve mathematical objects other than numbers. The logical values true and false can be combined using logic operations, such as and, or, and not. Vectors can be added and subtracted. [5] Rotations can be combined using the function composition operation, performing the first rotation and then the second. Operations on sets ...
[2] [3] Primitives of imperative programming languages rely on assignment to do iteration. [4] At the lowest level, assignment is implemented using machine operations such as MOVE or STORE. [2] [4] Variables are containers for values. It is possible to put a value into a variable and later replace it with a new one.
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 ]
Set operation may have one of the following meanings. Any operation with sets; Set operation (Boolean), Boolean set operations in the algebra of sets; Set operations (SQL), type of operation in SQL; Fuzzy set operations, a generalization of crisp sets for fuzzy sets