Equality, or more formally the identity relation, is the binary relation on a set X defined by .
The identity relation is the paradigmatic example of the more general concept of an equivalence relation on a set: those binary relations which are reflexive, symmetric, and transitive. The relation of equality is also antisymmetric. These four properties uniquely determine the equality relation on any set S and render equality the only relation on S that is both an equivalence relation and a partial order. It follows from this that equality is the smallest equivalence relation on any set S, in the sense that it is a subset of any other equivalence relation on S.
An equation is simply an assertion that two expressions are related by equality.
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Logical formulations
The equality relation is always defined such that things that are equal have all and only the same properties. Some people define equality as congruence. Often equality is just defined as identity.
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:
In this law, the connective "if and only if" can be weakened to "if"; the modified law is equivalent to the original.
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.
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, i.e. is well-formed).
In first-order logic, this is a schema, since we can't quantify over expressions like F (which would be a functional predicate).
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:
- For any quantity a, a = a.
This property is generally used in mathematical proofs as an intermediate step.
The symmetric property states:
The transitive property states:
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.
Although the symmetric and transitive properties are often seen as fundamental, they can be proved, if the substitution and reflexive properties are assumed instead.
References
- Mac Lane, Saunders; Garrett Birkhoff (1967). Algebra. American Mathematical Society.
See also
Categories: Elementary arithmetic | Identity
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