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In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. This defines a binary predicate, equality, denoted "="; x = y iff x and y are equal. Equivalence in the general sense is provided by the construction of a equivalence relation between two elements. A statement that two expressions denote equal quantities is an equation.
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.
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.
Leibniz's idea was that two things are identicalIn philosophy, identity is the quality of being "the same as". It is of particular interest to logicians and metaphysicians. Logic In logic, the identity relation is normally, (by definition), the transitive, symmetric, and reflexive relation that holds o if and only if they have precisely the same propertiesThis page deals with property as ownership rights. For information about property in the performing arts, see prop. For information about properties in philosophy, see property (philosophy Within the law, property is a general legal category for rights of. To formalise this, we wish to say
However, in first order logic, we cannot quantify over predicates. Thus, we need to use an axiom schema:
This axiom schema, valid for any predicate P in one variable, takes care of only one direction of Leibniz's law; if x and y are equal, then they have the same properties. We can take care of the other direction by simply postulating:
Then if x and y have the same properties, then in particular they are the same with respect to the predicate P given by P(z) iff x = z. Since P(x) holds, P(y) must also hold, so x = y.
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