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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]
The transitive property of inequality states that for any real numbers a, b, c: [8] If a ≤ b and b ≤ c , then a ≤ c . If either of the premises is a strict inequality, then the conclusion is a strict inequality:
More specifically, an equation represents a binary relation (i.e., a two-argument predicate) which may produce a truth value (true or false) from its arguments. In computer programming, the computation from the two expressions is known as comparison. [20] An equation can be used to define a set, called its solution set.
Reflexive and transitive: The relation ≤ on N. Or any preorder; Symmetric and transitive: The relation R on N, defined as aRb ↔ ab ≠ 0. Or any partial equivalence relation; Reflexive and symmetric: The relation R on Z, defined as aRb ↔ "a − b is divisible by at least one of 2 or 3." Or any dependency relation.
The first equation can be broken out into three equations, 1 ≤ a*, a*•a* ≤ a*, and a ≤ a*. Defining a to be reflexive when 1 ≤ a and transitive when a•a ≤ a by abstraction from binary relations, the first two of those equations force a* to be reflexive and transitive while the third forces a* to be greater or equal to a. The next ...
In mathematics, an inequation is a statement that an inequality holds between two values. [1] [2] It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific inequality relation.
All definitions tacitly require the homogeneous relation be transitive: for all ,,, if and then . A term's definition may require additional properties that are not listed in this table. In mathematics , a binary relation R {\displaystyle R} on a set X {\displaystyle X} is antisymmetric if there is no pair of distinct elements of X ...
Inequality (mathematics) ... Solving quadratic equations with continued fractions; T. Trairāśika; Transitive relation; Trinomial; Two-element Boolean algebra; U.