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  2. Transitive relation - Wikipedia

    en.wikipedia.org/wiki/Transitive_relation

    The transitive extension of R 1 would be denoted by R 2, and continuing in this way, in general, the transitive extension of R i would be R i + 1. The transitive closure of R, denoted by R* or R ∞ is the set union of R, R 1, R 2, ... . [8] The transitive closure of a relation is a transitive relation. [8]

  3. Relation (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Relation_(mathematics)

    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.

  4. Converse relation - Wikipedia

    en.wikipedia.org/wiki/Converse_relation

    In the monoid of binary endorelations on a set (with the binary operation on relations being the composition of relations), the converse relation does not satisfy the definition of an inverse from group theory, that is, if is an arbitrary relation on , then does not equal the identity relation on in general.

  5. Equivalence relation - Wikipedia

    en.wikipedia.org/wiki/Equivalence_relation

    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).

  6. Relation algebra - Wikipedia

    en.wikipedia.org/wiki/Relation_algebra

    A relation algebra (L, ∧, ∨, −, 0, 1, •, I, ˘) is an algebraic structure equipped with the Boolean operations of conjunction x∧y, disjunction x∨y, and negation x −, the Boolean constants 0 and 1, the relational operations of composition x•y and converse x˘, and the relational constant I, such that these operations and constants satisfy certain equations constituting an ...

  7. Law of trichotomy - Wikipedia

    en.wikipedia.org/wiki/Law_of_trichotomy

    A law of trichotomy on some set X of numbers usually expresses that some tacitly given ordering relation on X is a trichotomous one. An example is the law "For arbitrary real numbers x and y, exactly one of x < y, y < x, or x = y applies"; some authors even fix y to be zero, [1] relying on the real number's additive linearly ordered group structure.

  8. Inverse function - Wikipedia

    en.wikipedia.org/wiki/Inverse_function

    The square root of x is a partial inverse to f(x) = x 2. Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. For example, the function = is not one-to-one, since x 2 = (−x) 2.

  9. Intransitivity - Wikipedia

    en.wikipedia.org/wiki/Intransitivity

    This relation is intransitive since, for example, 2 R 6 (2 is a divisor of 6) and 6 R 3 (6 is a multiple of 3), but 2 is neither a multiple nor a divisor of 3. This does not imply that the relation is antitransitive (see below); for example, 2 R 6, 6 R 12, and 2 R 12 as well. An example in biology comes from the food chain.