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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 ...
Intersections of the unaccented modern Greek, Latin, and Cyrillic scripts, considering only the shapes of the letters and ignoring their pronunciation Example of an intersection with sets The intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B {\displaystyle A\cap B} , [ 3 ] is the set of all objects that ...
The combined region of the two sets is called their union, denoted by A ∪ B, where A is the orange circle and B the blue. The union in this case contains all living creatures that either are two-legged or can fly (or both). The region included in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B.
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.
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 ...
For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B.
For example, ≤ is the union of < and =, and ≥ is the union of > and =. Intersection [e] If R and S are relations over X then R ∩ S = { (x, y) | xRy and xSy} is the intersection relation of R and S. The identity element of this operation is the universal relation. For example, "is a lower card of the same suit as" is the intersection of ...
The situation that appears in the derangement example above occurs often enough to merit special attention. [7] Namely, when the size of the intersection sets appearing in the formulas for the principle of inclusion–exclusion depend only on the number of sets in the intersections and not on which sets appear. More formally, if the intersection