Reference
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.[1][2] The symbol "=" is called an "equals sign". Two objects that are not equal are said to be distinct.
For example:
{\displaystyle x=y}x=y means that x and y denote the same object.[3]
The identity {\displaystyle (x+1)^{2}=x^{2}+2x+1}{\displaystyle (x+1)^{2}=x^{2}+2x+1} 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.
{\displaystyle \{x\mid P(x)\}=\{x\mid Q(x)\}}{\displaystyle \{x\mid P(x)\}=\{x\mid Q(x)\}} if and only if {\displaystyle P(x)\Leftrightarrow Q(x).}{\displaystyle P(x)\Leftrightarrow Q(x).} This assertion, which uses set-builder notation, means that if the elements satisfying the property {\displaystyle P(x)}P(x) are the same as the elements satisfying {\displaystyle Q(x),}{\displaystyle Q(x),} 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.[4]
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Basic properties Substitution property: For any quantities a and b and any expression F(x), if a = b, then F(a) = F(b) (provided that both sides are well-formed). 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).
Reflexive property: For any quantity a, a = a. Symmetric property: For any quantities a and b, if a = b, then b = a. Transitive property: For any quantities a, b, and c, if a = b and b = c, then a = c.
These 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.
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Relation with equivalence and isomorphism
Main articles: Equivalence relation and Isomorphism
Viewed as a relation, equality is the archetype of the more general concept of an equivalence relation on a set: those binary relations that are reflexive, symmetric and transitive. The identity relation is an equivalence relation. Conversely, let R be an equivalence relation, and let us denote by xR the equivalence class of x, consisting of all elements z such that x R z. Then the relation x R y is equivalent with the equality xR = yR. It follows that equality is the finest equivalence relation on any set S in the sense that it is the relation that has the smallest equivalence classes (every class is reduced to a single element).
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In some contexts, equality is sharply distinguished from equivalence or isomorphism.
For example, one may distinguish fractions from rational numbers, the latter being equivalence classes of fractions: the fractions "1/2 "and "2/4" are distinct as fractions (as different strings of symbols) but they "represent" the same rational number (the same point on a number line). This distinction gives rise to the notion of a quotient set.
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Similarly, the sets
{\displaystyle \{{\text{A}},{\text{B}},{\text{C}}\}}{\displaystyle \{{\text{A}},{\text{B}},{\text{C}}\}} and {\displaystyle \{1,2,3\}}{\displaystyle \{1,2,3\}} are not equal sets — the first consists of letters, while the second consists of numbers — but they are both sets of three elements and thus isomorphic, meaning that there is a bijection between them. For example
{\displaystyle {\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.}\text{A} \mapsto 1, \text{B} \mapsto 2, \text{C} \mapsto 3. However, there are other choices of isomorphism, such as
{\displaystyle {\text{A}}\mapsto 3,{\text{B}}\mapsto 2,{\text{C}}\mapsto 1,}\text{A} \mapsto 3, \text{B} \mapsto 2, \text{C} \mapsto 1, and these sets cannot be identified without making such a choice — any statement that identifies them "depends on choice of identification". This distinction, between equality and isomorphism, is of fundamental importance in category theory and is one motivation for the development of category theory.
https://en.wikipedia.org/wiki/Equivalence_relation
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https://en.wikipedia.org/wiki/Equality_(mathematics)
