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Fundamentals. 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 simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement (postfix ') of sets. These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}.
Three sets involved. [edit] In the left hand sides of the following identities, L{\displaystyle L}is the L eft most set, M{\displaystyle M}is the M iddle set, and R{\displaystyle R}is the R ight most set. Precedence rules. There is no universal agreement on the order of precedenceof the basic set operators.
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.
De Morgan's Laws represented as a circuit with logic gates (International Electrotechnical Commission diagrams). In extensions of classical propositional logic, the duality still holds (that is, to any logical operator one can always find its dual), since in the presence of the identities governing negation, one may always introduce an operator ...
Statement. The symmetric difference is the set of elements that are in either set, but not in the intersection. Symbolic statement. A Δ B = ( A ∖ B ) ∪ ( B ∖ A ) {\displaystyle A\,\Delta \,B=\left (A\setminus B\right)\cup \left (B\setminus A\right)} In mathematics, the symmetric difference of two sets, also known as the disjunctive union ...
Venn diagram showing the union of sets A and B as everything not in white. In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as
Set theory. Statement. The intersection of A and B is the set A ∩ B of elements that lie in both set A and set B . Symbolic statement. A ∩ B = {x: x ∈ A and x ∈ B} In set theory, the intersection of two sets and denoted by 1 is the set containing all elements of that also belong to or equivalently, all elements of that also belong to 2.