Equivalence relation: Difference between revisions

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:<math>x_1 \stackrel{f}{\equiv} x_2 \Leftrightarrow f(x_1) = f(x_2) . \,</math>
:<math>x_1 \stackrel{f}{\equiv} x_2 \Leftrightarrow f(x_1) = f(x_2) . \,</math>


This is an equivalence relation, the '''[[Kernel of a function|kernel]]''' of ''f''.  Every equivalence relation can be defined in this say for a suitable function ''f''.
This is an equivalence relation, the '''[[Kernel of a function|kernel]]''' of ''f''.  Every equivalence relation can be defined in this way for a suitable function ''f''.

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In mathematics, an equivalence relation is a relation between that formalises the intuitive notion that related objects have some property in common. Examples include equality between numbers or other quantities; geometrical relations such as parallel, congruence, similarity or between-ness; abstract algebraic concepts such as isomorphism or homeomorphism. The equivalence may be expressed by formulae, geometric concepts or algorithms, but in keeping with the modern definition of mathematics, it is most convenient to identify an equvialence relation with the sets of objects for which it holds true.

A relation on a set X is a equivalence relation if it satisfies the following three properties

  • is reflexive: for all .
  • is symmetric: .
  • is transitive: .

An equivalence class for is the set of elements of X all related to some particular element

The equivalence classes for a partition of the set X, that is, two classes and are either equal (have the same members), which is the case when , or are disjoint (have no members in common), which is the case when .

Examples

  • On any set X, the identity relation . The equivalence classes are the singleton sets .
  • On any set X, the universal relation for all x,y. There is one equivalence class, X itself.
  • On the integers, parity: iff m,n have the same remainder on division by 2. There are two equivalence classes, "even" and "odd".
  • On the integers more generally, modular arithmetic operates on the equivalence classes defined by remainder on division my a fixed modulus M.
  • On lines in the plane, being parallel. The equivalence classes are the directions.
  • On triangles in the plane, being congruent.
  • On triangles in the plane, being similar.

Quotient classes

It will be seen in the examples that a common way of defining an equivalence relation is to state that elements have some common property. We can formalise this by saying that if f is a function defined on the set X, we define the relation by

This is an equivalence relation, the kernel of f. Every equivalence relation can be defined in this way for a suitable function f.