Two mathematical objects are equal if and only if they are precisely the same in every way. This defines a binary relation, equality, denoted by the sign of equality "=" in such a way that the statement "x = y" means that x and y are equal. Image File history File links Eqaulity. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... The equal sign, equals sign, or = is a mathematical symbol used to indicate equality. ...

Equivalence in a more general sense is provided by the construction of an equivalence relation between two sets. A statement that two expressions denote equal quantities is an equation. In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... An expression in the very basic sense is the noun form of the verb express. ... In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ...

Beware that sometimes a statement of the form "A = B" may not be an equality. For example, the statement T(n) = O(n²) means that T(n) grows at the order of n². It is not an equality, because the sign "=" in the statement is not the equality sign; indeed, it is meaningless to write O(n²) = T(n). See Big O notation for more on this. It has been suggested that Landau notation be merged into this article or section. ...

Given a set A, the restriction of equality to the set A is a binary relation, which is at once reflexive, symmetric, antisymmetric, and transitive. Indeed it is the only relation on A with all these properties. Dropping the requirement of antisymmetry yields the notion of equivalence relation. Conversely, given any equivalence relation R, we can form the quotient set A/R, and the equivalence relation will 'descend' to equality in A/R. Note that it may be impractical to compute with equivalence classes: one solution often used is to look for a distinguished normal form representative of a class. In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity. ... In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a. ... In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets... The term normal form is used in a variety of contexts. ...

Logical formulations

The equality relation is always defined such that things that are equal have all and only the same properties. Often equality is just defined as identity. In philosophy, identity is whatever makes an entity definable and recognizable, in terms of possessing a set of qualities or characteristics that distinguish it from entities of a different type. ...

A stronger sense of equality is obtained if some form of Leibniz's law is added as an axiom; the assertion of this axiom rules out "bare particulars"—things that have all and only the same properties but are not equal to each other—which are possible in some logical formalisms. The axiom states that two things are equal if they have all and only the same properties. Formally: The identity of indiscernibles, also known as Leibnizs Law, is an ontological principle first forumlated by German philosopher Göttfried Wilhelm Leibniz. ... An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. ... // Use of the term In common usage, property means ones own thing and refers to the relationship between individuals and the objects which they see as being their own to dispense with as they see fit. ...

Given any x and y, x = y if, given any predicate P, P(x) if and only if P(y).

In this law, the connective "if and only if" can be weakened to "if"; the modified law is equivalent to the original. In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... In mathematics, a predicate is a relation. ...

Instead of considering Leibniz's law as an axiom, it can also be taken as the definition of equality. The property of being an equivalence relation, as well as the properties given below, can then be proved: they become theorems. A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...

Some basic logical properties of equality

The substitution property states:

• For any quantities a and b and any expression F(x), if a = b, then F(a) = F(b) (if either side makes sense).

In first-order logic, this is a schema, since we can't quantify over expressions like F (which would be a functional predicate). In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... It has been suggested that Predicate calculus be merged into this article or section. ... The word schema comes from the Greek word σχήμα (skhēma) that means shape or more generally plan. ... In formal logic and related branches of mathematics, a functional predicate, or function symbol, is a logical symbol that may be applied to an object term to produce another object term. ...

Some specific examples of this are:

• For any real numbers a, b, and c, if a = b, then a + c = b + c (here F(x) is x + c);
• For any real numbers a, b, and c, if a = b, then a − c = b − c (here F(x) is x − c);
• For any real numbers a, b, and c, if a = b, then ac = bc (here F(x) is xc);
• For any real numbers a, b, and c, if a = b and c is not zero, then a/c = b/c (here F(x) is x/c).

The reflexive property states: In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... 0 (zero) is both a number and a numeral. ...

For any quantity a, a = a.

This property is generally used in mathematical proofs as an intermediate step. In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...

The symmetric property states:

• For any quantities a and b, if a = b, then b = a.

The transitive property states: In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...

• For any quantities a, b, and c, if a = b and b = c, then a = c.

The binary relation "is approximately equal" between real numbers or other things, even if more precisely defined, is not transitive (it may seem so at first sight, but many small differences can add up to something big). However, equality almost everywhere is transitive. In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... Logical conjunction (usual symbol and) is a logical operator that results in true if both of the operands are true. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... An approximation is an inexact representation of something that is still close enough to be useful. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Difference is the contrary of equality, in particular of objects. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...

Although the symmetric and transitive properties are often seen as fundamental, they can be proved from the substitution and reflexive properties.

See also

• Equals sign
• Equation