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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.
De Morgan's laws represented with Venn diagrams.In each case, the resultant set is the set of all points in any shade of blue. In propositional logic and Boolean algebra, De Morgan's laws, [1] [2] [3] also known as De Morgan's theorem, [4] are a pair of transformation rules that are both valid rules of inference.
An element x is called a dual distributive element if ∀y,z: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). In a distributive lattice, every element is of course both distributive and dual distributive. In a non-distributive lattice, there may be elements that are distributive, but not dual distributive (and vice versa).
A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science.
These properties can be visualized with Venn diagrams. They also follow from the fact that P(U) is a Boolean lattice. The properties followed by "L" interpret the lattice axioms. Elementary discrete mathematics courses sometimes leave students with the impression that the subject matter of set theory is no more than these properties.
The principle of distributivity states that the algebraic distributive law is valid, where both logical conjunction and logical disjunction are distributive over each other so that for any propositions A, B and C the equivalences
Hasse diagram of the Boolean algebra of divisors of 30. For any natural number n, the set of all positive divisors of n, defining a ≤ b if a divides b, forms a distributive lattice. This lattice is a Boolean algebra if and only if n is square-free. The bottom and the top elements of this Boolean algebra are the natural numbers 1 and n ...
In this case, if the choice of U is clear from the context, the notation A c is sometimes used instead of U \ A, particularly if U is a universal set as in the study of Venn diagrams. Symmetric difference of sets A and B , denoted A B or A ⊖ B , is the set of all objects that are a member of exactly one of A and B (elements which are in one ...