Ads
related to: transitive relation set example worksheet high school pdf free
Search results
Results From The WOW.Com Content Network
In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c. Every partial order and every equivalence relation is transitive. For example, less than and equality among real numbers are both transitive: If a < b and b < c then a < c; and if x ...
In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [ 1 ] 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 ...
Note that this is the set of all of the objects related to by the transitive closure of the membership relation, since the union of a set can be expressed in terms of the relative product of the membership relation with itself. The transitive closure of a set can be expressed by a first-order formula: is a transitive closure of iff is an ...
Transitive closure. In mathematics, the transitive closure R+ of a homogeneous binary relation R on a set X is the smallest relation on X that contains R and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets R+ is the unique minimal transitive superset of R.
Fundamentals. The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
Quasitransitive relation. The quasitransitive relation ≤ 4. Its symmetric and transitive part is shown in blue and green, respectively. The mathematical notion of quasitransitivity is a weakened version of transitivity that is used in social choice theory and microeconomics. Informally, a relation is quasitransitive if it is symmetric for ...
Partial orders. A reflexive, weak, [1] or non-strict partial order, [2] commonly referred to simply as a partial order, is a homogeneous relation ≤ on a set that is reflexive, antisymmetric, and transitive. That is, for all it must satisfy: Reflexivity: , i.e. every element is related to itself.
The Mostowski collapse lemma states that for every such R there exists a unique transitive class (possibly proper) whose structure under the membership relation is isomorphic to (X, R), and the isomorphism is unique. The isomorphism maps each element x of X to the set of images of elements y of X such that y R x (Jech 2003:69).