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Concept
Equality (mathematics)
Summary
In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality between A and B is written A = B, and pronounced "A equals B". The symbol "=" is called an "equals sign". Two objects that are not equal are said to be distinct.
For example:
means that x and y denote the same object.
The identity means that if x is any number, then the two expressions have the same value. This may also be interpreted as saying that the two sides of the equals sign represent the same function.
if and only if This assertion, which uses set-builder notation, means that if the elements satisfying the property are the same as the elements satisfying then the two uses of the set-builder notation define the same set. This property is often expressed as "two sets that have the same elements are equal." It is one of the usual axioms of set theory, called axiom of extensionality.
The etymology of the word is from the Latin aequālis ("equal", "like", "comparable", "similar") from aequus ("equal", "level", "fair", "just").
These last three properties make equality an equivalence relation. They were originally included among the Peano axioms for natural numbers. Although the symmetric and transitive properties are often seen as fundamental, they can be deduced from substitution and reflexive properties.
When A and B are not fully specified or depend on some variables, equality is a proposition, which may be true for some values and false for other values. Equality is a binary relation (i.e., a two-argument predicate) which may produce a truth value (false or true) from its arguments. In computer programming, its computation from the two expressions is known as comparison.
Identity (mathematics)
When A and B may be viewed as functions of some variables, then A = B means that A and B define the same function. Such an equality of functions is sometimes called an identity.
Official source
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