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If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the ...
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 ...
A coanalytic set is the complement of an analytic set cofinal A subset of a poset is called cofinal if every element of the poset is at most some element of the subset. cof cofinality cofinality 1. The cofinality of a poset (especially an ordinal or cardinal) is the smallest cardinality of a cofinal subset 2.
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 power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see Cardinality of the continuum). The power set of a set S, together with the operations of union, intersection and complement, is a Σ-algebra over S and can be viewed as the prototypical example of a Boolean algebra.
A relation is connected if, and only if, its complement is anti-symmetric. A relation is strongly connected if, and only if, its complement is asymmetric. [21] If R and S are relations over a set X, and R is contained in S, then If R is reflexive, connected, strongly connected, left-total, or right-total, then so is S.
Pascal's calculator had two sets of result digits, a black set displaying the normal result and a red set displaying the nines' complement of this. A horizontal slat was used to cover up one of these sets, exposing the other. To subtract, the red digits were exposed and set to 0. Then the nines' complement of the minuend was entered.