Beware that sometimes a statement of the form "A = B" may not be an equality. For example, the statement T(n) = O(n^{2}) means that T(n) grows at the order of n^{2}. It is not an equality, because the sign "=" in the statement is not the equality sign; indeed, it is meaningless to write O(n^{2}) = 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 revert to equality in A/R.
Predicate logic contains standard axioms for equality that formalise Leibniz's law[?], put forward by the philosopher Gottfried Leibniz in the 1600s. Leibniz's idea was that two things are identical if and only if they have precisely the same properties. To formalise this, we wish to say
Some basic logical properties of equality
Although the symmetric and transitive properties are often seen as fundamental, they can be proved from the substitution and reflexive properties.
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