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Every two-element set serves as a subobject classifier in Set. The power object of a set A is given by its power set, and the exponential object of the sets A and B is given by the set of all functions from A to B. Set is thus a topos (and in particular cartesian closed and exact in the sense of Barr). Set is not abelian, additive nor preadditive.
Similarly, every finite Boolean algebra can be represented as a power set – the power set of its set of atoms; each element of the Boolean algebra corresponds to the set of atoms below it (the join of which is the element). This power set representation can be constructed more generally for any complete atomic Boolean algebra.
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.
Define the measure of a set to be its cardinality, that is, the number of elements in the set. Then, each of the singletons {i}, for i = 1, 2, ..., 9, 10 is an atom. Consider the Lebesgue measure on the real line. This measure has no atoms.
Every finite partially ordered set with 0 is atomic, but the set of nonnegative real numbers (ordered in the usual way) is not atomic (and in fact has no atoms). A partially ordered set is relatively atomic (or strongly atomic) if for all a < b there is an element c such that a <: c ≤ b or, equivalently, if every interval [a, b] is atomic ...
Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.' [1] Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the ...
is_sub_bag(B 1, B 2): checks whether each element in the bag B 1 occurs in B 1 no more often than it occurs in the bag B 2; sometimes denoted as B 1 ⊑ B 2. count(B, x): returns the number of times that the element x occurs in the bag B; sometimes denoted as B # x. scaled_by(B, n): given a natural number n, returns a bag which contains the ...
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".