Search results
Results From The WOW.Com Content Network
Equivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number.
A relation is said to be coanalytic if its complement is an analytic set. Silver's dichotomy is a statement about the equivalence classes of a coanalytic equivalence relation, stating any coanalytic equivalence relation either has countably many equivalence classes, or else there is a perfect set of reals that are each incomparable to each other. [4]
For any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from any partition P of X, we can define an equivalence relation on X by setting x ~ y precisely when x and y are in the same part in P. Thus the notions of equivalence relation and partition are essentially equivalent. [5]
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 ...
Equivalence relation ... This set is partially, but not totally, ordered because there is a relationship from 1 to every other number, but there is no relationship ...
Let be the set of ordered pairs of integers (,) with non-zero , and define an equivalence relation on such that (,) (,) if and only if =, then the equivalence class of the pair (,) can be identified with the rational number /, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of ...
The equivalence relation of equality is a special case, as, if restricted to a given set , it is the strictest possible equivalence relation on ; specifically, equality partitions a set into equivalence classes consisting of all singleton sets.
The signature of equivalence relations has one binary infix relation symbol ~, no constants, and no functions. Equivalence relations satisfy the axioms: Reflexive ∀x x~x; Symmetric ∀x ∀y x~y → y~x; Transitive: ∀x ∀y ∀z (x~y ∧ y~z) → x~z. Some first-order properties of equivalence relations are: ~ has an infinite number of ...