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In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, 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 ...
In mathematical logic, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set theory.
A binary operation ∗ on the set S is associative when this diagram commutes. That is, when the two paths from S×S×S to S compose to the same function from S×S×S to S. Formally, a binary operation on a set S is called associative if it satisfies the associative law:
A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }. [5] Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets. A derived binary relation between two sets is the subset relation, also called set inclusion.
The axiom schema of replacement asserts that the image of a set under any definable function will also fall inside a set. Formally, let φ {\displaystyle \varphi } be any formula in the language of ZFC whose free variables are among x , y , A , w 1 , … , w n , {\displaystyle x,y,A,w_{1},\dotsc ,w_{n},} so that in particular B {\displaystyle B ...
8.3.3 Exact formulas/equalities for images of set operations. 8.3.3.1 Formulas for f ... Download as PDF; Printable version; ... show that it is a Boolean algebra. ...
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