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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.
Quasi-set theory; Relation; Rough set; Russell's paradox; Semiset; Set theory. Alternative set theory; Axiomatic set theory; General set theory; Kripke–Platek set theory with urelements; Morse–Kelley set theory; Naive set theory; New Foundations; Pocket set theory; Positive set theory; S (Boolos 1989) Scott–Potter set theory; Tarski ...
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.
It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.
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 ...
As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the values 3 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false.
Some perspectives contrast ethics and value theory, asserting that the normative concepts examined by ethics are distinct from the evaluative concepts examined by value theory. [21] Axiological ethics is a subfield of ethics examining the nature and role of values from a moral perspective, with particular interest in determining which ends are ...
It can be shown that the B-valued relations ‖ ∈ ‖ and ‖ = ‖ on V B make V B into a Boolean-valued model of set theory. Each sentence of first-order set theory with no free variables has a truth value in B; it must be shown that the axioms for equality and all the axioms of ZF set theory (written without free variables) have truth ...