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The equals sign, used to represent equality symbolically in an equation. In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object.
Being characterized by a universal property is the standard feature of free objects in universal algebra. In the language of category theory , the construction of the free group (similar to most constructions of free objects) is a functor from the category of sets to the category of groups .
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 .
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.
Elementary algebra, also known as high school algebra or college algebra, [1] encompasses the basic concepts of algebra. It is often contrasted with arithmetic : arithmetic deals with specified numbers , [ 2 ] whilst algebra introduces variables (quantities without fixed values).
In mathematics, a property is any characteristic that applies to a given set. [1] Rigorously, a property p defined for all elements of a set X is usually defined as a function p: X → {true, false}, that is true whenever the property holds; or, equivalently, as the subset of X for which p holds; i.e. the set {x | p(x) = true}; p is its indicator function.
For example: For all real numbers a and b, if a = b, then a ≥ 0 implies b ≥ 0 (here, () is x ≥ 0). This is a property which is most often used in algebra, especially in solving systems of equations, but is apllied in nearly every area of math that uses equality
Algebra is one of the main branches of mathematics, covering the study of structure, relation and quantity. Algebra studies the effects of adding and multiplying numbers , variables , and polynomials , along with their factorization and determining their roots .
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