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In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object. [1] [2] Equality between A and B is written A = B, and pronounced "A equals B". In this equality, A and B are distinguished by calling them left-hand side (LHS), and right-hand side ...
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.
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".
Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality , which is concerned with whether the internal definitions of objects are the same.
Natural numbers are also used as labels, like jersey numbers on a sports team, where they serve as nominal numbers and do not have mathematical properties. [5] The natural numbers form a set, commonly symbolized as a bold N or blackboard bold . Many other number sets are built from the natural numbers.
The Egyptians used the commutative property of multiplication to simplify computing products. [7] [8] Euclid is known to have assumed the commutative property of multiplication in his book Elements. [9] Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of ...
The pure theory of equality contains formulas of first-order logic with equality, where the only predicate symbol is equality itself and there are no function symbols. Consequently, the only form of an atomic formula is x = y {\displaystyle x=y} where x , y {\displaystyle x,y} are (possibly identical) variables.