Equality (mathematics): definition, properties, notation and examples
Mathematical equality is the fundamental relation stating two expressions denote the same object or value. This article explains formal properties, notation, variants, applications, and common distinctions.
Overview
In mathematics, equality is the basic binary relation that asserts two expressions represent the same mathematical object or value. When we write an expression of the form x = y we claim that x and y are interchangeable in every context where equality is meaningful. The notion is central to arithmetic, algebra, logic, set theory and many branches of applied mathematics. For a concise entry point, see mathematical equality, and for a general description of what it means for two things to be 'the same' in mathematics see identity. The symbol '=' is the most common notation; its origin and conventional use are discussed below, and the glyph itself is treated as a topic of historical interest here.
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Formal properties
Equality is not arbitrary: it is commonly taken to satisfy a small set of structural properties that make it an equivalence relation and a tool for substitution. The three standard algebraic properties are:
- Reflexivity: every object equals itself (for all x, x = x).
- Symmetry: if x = y then y = x.
- Transitivity: if x = y and y = z then x = z; this property is often highlighted in elementary logical expositions here.
Beyond these, equality satisfies the substitution property (sometimes called Leibniz's law): if x = y, then any property or formula that holds of x also holds of y. In formal logic and axiomatic systems, identity is usually introduced by axioms that encode these behaviors so that equality can be used to replace equals by equals in proofs and calculations.
Notation, equations and inequalities
The expression that two quantities are equal is called an equation or an equality; equations are solved, manipulated and used to define relationships among variables (equations). By contrast, relations that compare sizes or orderings—such as <, >, ≤ or ≥—are inequalities and express non-equality relationships. Other notations convey related ideas: a double bar or triple bar (≡) can denote congruence of residue classes or definitional identity, while the tilde (~) is commonly used for asymptotic or equivalence relations that are not strict identity.
Variants: equivalence relations and geometric congruence
When two objects are considered the same only up to some prescribed transformation, mathematicians use the more general concept of an equivalence relation. An equivalence relation is any relation that is reflexive, symmetric and transitive; it partitions a collection of objects into classes of mutually equivalent items. Standard examples include congruence classes in modular arithmetic and similarity classes of geometric figures. For definitions and examples see equivalence relations. In geometry, the term congruence describes when two figures match under rigid motions (translations, rotations, reflections) while similarity allows uniform scaling; these ideas are discussed in elementary geometry sources (geometry), with congruence usually symbolized by ≅ (congruence) and similarity by ~ (similarity).
Equality in logic, set theory and mathematics of structure
In formal systems, equality is handled carefully. In first-order logic with identity, special axioms govern how identity interacts with predicates and functions, and in set theory equality is tied to extensionality: two sets are equal exactly when they have the same elements. Structural or abstract equality can mean different things depending on context — for example two functions may be considered equal if they have the same domain and assign the same value to each input, while two algebraic structures might be equal only if there is an identity-preserving bijection between them. These distinctions underlie much of modern algebra and category theory, where 'sameness' is often relaxed to isomorphism or equivalence rather than strict elementwise identity.
Equality in computation and programming
Computer science applies the mathematical idea but must also distinguish between value equality and reference or identity. Many programming languages provide an operator for comparing values and a separate notion for checking whether two variables refer to the same object in memory. Languages influenced by object-oriented design address this distinction explicitly; for a general overview see equality in computing and a discussion specific to object-oriented contexts here. Practical issues arise with numeric approximations (for example floating-point comparisons), immutable versus mutable objects, and operator overloading. Some languages introduce separate operators (e.g. ==, === or an equals method) to make the semantic difference explicit.
Uses, examples and notable distinctions
Equality is used everywhere in mathematics: to state identities like trigonometric equalities, to define functions and relations, to specify axioms, and to reduce expressions during computation. Concrete examples include the identity 2 + 2 = 4, the algebraic equality (x+1)^2 = x^2 + 2x + 1, and the set-theoretic equality {a,b} = {b,a}. Important distinctions to bear in mind are:
- Strict equality versus equivalence up to a relation (isomorphism, congruence, similarity).
- Mathematical identity versus computational identity (value vs reference).
- Exact equality versus approximate equality used in numerical analysis.
For historical notes about the adoption of the equals sign and for further reading see notation, and for social or metaphorical uses of equality outside mathematics consult treatments in the social sciences here (for example how people identify peers peer).
Further reading
To explore related terms and applications consult introductory texts on algebra and logic, entries about equations, the study of equivalence relations, practical programming guides on equality semantics (computing), and elementary geometry references on congruence and similarity.
Equality within a set or structure
Mathematics deals with the relationship between mathematical objects within a set provided with a mathematical structure, but not with the nature of a mathematical object independent of the sets and structures to which it belongs. Therefore, it is a meaningful question in mathematics whether two objects from different sets are the same or different from each other only if one set is part of the other or a set superior to both is needed. For example, whether the cardinal number 3 (in the sense of set theory: the power of a three-element set) is the same object as the real number 3 is only interesting if one wants to build a structure in which cardinal numbers occur next to real numbers in the same context - an unusual case in which one has to define how the equality is meant.
But if one or more sets are defined unambiguously, it is clear what equality means: the elements of a set are equal only to themselves, and two sets are equal if they contain the same elements. Based on this, one can form pairs and n-tuples using the Cartesian product of sets, as well as functions that map a set into itself or into another set. Equality transfers to such composite objects, where equal is what is built up in the same way from the same components.
As an example, consider the construction of the set 
of the rational numbers from the set 
of the integers serve. Rational numbers are, at first sight, fractions of integers with nonzero denominators, conceived as a set of pairs of numbers from 
. But then the pairs ⟨ 
and ⟨ would be 
two different pairs, hence, according to the definition of equality, two different rational numbers 
and 
, and a definitional specification that they should be equal would lead to a contradiction. How one nevertheless arrives at a set of pairwise different rational numbers by forming equivalence classes is described in detail in the Definition section of the article Rational Number.
Statements about the rational numbers defined in such a way can be made only if functions are defined there as for example the arithmetic operations or relations as for example the smaller and larger relation. If one does not have this, there are at most statements about the equality of two differently written rational numbers, whose correctness or incorrectness is already determined on the basis of the definition of the rational numbers. In other words, equality is indeed a relation on the set of the rational numbers, but not one that could have been defined there after the definition of could still have been defined there. Rather, it arose from the definition of came into being with.
Let us take as an example of a statement about rational numbers the equation 
. It only makes sense if it is known,
- which set of objects (here the set of rational numbers) we are talking about,
- how the arithmetic operations (here the four basic arithmetic operations - exponentiation with constant natural numbers is only an abbreviated notation for repeated multiplications) are defined on this basic set of objects and
- for which elements of the set the occurring free variables (here the
and the 
) stand.
These three things, namely the underlying set of objects, the definition of the occurring functions and relations on this set - but not that of the relation equality - as well as the occupation of the free variables with elements of the set thus form the context in which the statement is interpreted, i.e. is true or false in an unambiguous way. This can be done in a formal way as shown under Interpretation, but even without such formalism every statement has a meaning only if these three components of the interpretation of the statement are fixed.
The assignment of the free variables usually results from the context. In this example - as with all formulas found in formula collections - it is usually meant that the statement is generally valid, i.e. it applies to all and from the basic set. In another context, the formula could have represented the task of finding all for given such that the equation is satisfied (see Equation). Equations and other statements containing free variables about which nothing is specified in the context of use may be generally valid, satisfiable, or unsatisfiable, depending on whether they are true for all, some, or no assignments of the free variables to elements of the basic set.
The other two components of the interpretation, i.e. the basic set and the functions and relations defined on it, together form the mathematical structure in the context of which the statement is generally valid, satisfiable or unsatisfiable. In the structure consisting of with the basic arithmetic operations, the said equation is universally valid; in structures with noncommutative multiplication it is not, for example, in the 2x2 matrices of integers with the usual matrix multiplication.
Questions and answers
Q: What is the symbol used to represent equality in mathematics?
A: The equals sign (=) is used to represent equality in mathematics.
Q: How can two mathematical objects be equivalent?
A: Two mathematical objects can be equivalent if they are related by an equivalence relation. This is often represented using symbols such as ∼ or ≡.
Q: What does it mean when two expressions denote equal quantities?
A: When two expressions denote equal quantities, it means that they are equal and this statement is referred to as an equation or an equality.
Q: How do mathematicians differentiate between equations and inequalities?
A: Equations are equal while inequalities are unequal.
Q: What is the difference between congruence and similarity in geometry?
A: Congruence occurs when one geometrical object can be moved or rotated so that it fits exactly where the other one is, without shrinking or enlarging either of them. Similarity occurs when shrinking or enlarging one of the two objects is needed for them to fit together. The congruence relation is often represented by the symbol ≅ while the similarity relation is represented by the symbol ∼ .
Q: In computer science, what operator compares actual values of objects rather than where variables point to?
A: In computer science, languages which have pointers usually use another operator (such as Java's 'equals' method) which compares actual values of objects rather than where variables point to.
Q: How do people define equality in social sciences?
A: In social sciences, two people are considered equal if many of the same things are true about them, such as having similar levels of education and money and being around the same age. Another name for a person who is equal to another person in this sense would be a peer.
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AlegsaOnline.com Equality (mathematics): definition, properties, notation and examples Leandro Alegsa
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