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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]
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.
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.
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 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 ...
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.
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.
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.