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In set theory, Scott's trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65) by referring to levels of the cumulative hierarchy. The method relies on the axiom of regularity but not on the axiom of choice .
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
In general, the join and meet of a subset of a partially ordered set need not exist. Join and meet are dual to one another with respect to order inversion. A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice.
For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1. The empty relation R (defined so that aRb is never true) on a set X is vacuously symmetric and transitive; however, it is not reflexive (unless X itself is empty).
A main area of study in invariant descriptive set theory is the relative complexity of equivalence relations. An equivalence relation on a set is considered more complex than an equivalence relation on a set if one can "compute using " - formally, if there is a function : which is well behaved in some sense (for example, one often requires that is Borel measurable) such that ,: ().
There are four different equivalence relations which may be defined on the set of functions f from N to X: equality; equality up to a permutation of N; equality up to a permutation of X; equality up to permutations of N and X. The three conditions on the functions and the four equivalence relations can be paired in 3 × 4 = 12 ways.
The cardinality of a set X is essentially a measure of the number of elements of the set. [1] Equinumerosity has the characteristic properties of an equivalence relation (reflexivity, symmetry, and transitivity): [1] Reflexivity Given a set A, the identity function on A is a bijection from A to itself, showing that every set A is equinumerous ...
As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the values 3 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false.