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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 ...
Given a set X, a relation R over X is a set of ordered pairs of elements from X, formally: R ⊆ { (x,y) | x, y ∈ X}. [2] [10] The statement (x,y) ∈ R reads "x is R-related to y" and is written in infix notation as xRy. [7] [8] The order of the elements is important; if x ≠ y then yRx can be true or false independently of xRy.
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.
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.
Equivalently, a comparability graph is a graph that has a transitive orientation, [3] an assignment of directions to the edges of the graph (i.e. an orientation of the graph) such that the adjacency relation of the resulting directed graph is transitive: whenever there exist directed edges (x,y) and (y,z), there must exist an edge (x,z).
Every connected symmetric graph must be vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree, [2] so that half-transitive graphs of odd degree do not exist. However, there do exist half-transitive graphs of even degree. [3] The smallest half-transitive graph is the Holt graph, with degree 4 and 27 vertices ...
Vertex-transitive graph, a graph whose automorphism group acts transitively upon its vertices Transitive set a set A such that whenever x ∈ A , and y ∈ x , then y ∈ A Topological transitivity property of a continuous map for which every open subset U' of the phase space intersects every other open subset V , when going along trajectory
At =, however, there is a problem: the graph of the square root function becomes vertical, corresponding to a horizontal tangent for the square function. = (for real x) has inverse = (for positive )