Equality (mathematics)
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.
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