Likhetsrelasjoner

Likhetsrelasjoner, also known as equivalence relations, are a fundamental concept in mathematics and computer science. A binary relation R on a set A is a likhetsrelasjon if it satisfies three properties: reflexivity, symmetry, and transitivity.

Reflexivity means that every element in the set A is related to itself. For any element a in A, the relation (a, a) must hold. For example, "is the same age as" is reflexive because every person is the same age as themselves.

Symmetry means that if an element a is related to an element b, then element b must also be related to element a. If (a, b) is in R, then (b, a) must also be in R. An example of a symmetric relation is "is married to". If person A is married to person B, then person B is married to person A.

Transitivity means that if an element a is related to an element b, and element b is related to an element c, then element a must also be related to element c. If (a, b) is in R and (b, c) is in R, then (a, c) must also be in R. The relation "is taller than or equal to" is transitive. If A is taller than or equal to B, and B is taller than or equal to C, then A is taller than or equal to C.

When a relation is a likhetsrelasjon, it partitions the set A into disjoint subsets called equivalence classes. All elements within an equivalence class are related to each other, and no element in one equivalence class is related to any element in another. This partitioning property makes likhetsrelasjoner very useful for grouping and categorizing objects based on shared characteristics. Examples of likhetsrelasjoner include equality, congruence in geometry, and equivalence modulo n in number theory.