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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 ...
The reason is that properties defined by bounded formulas are absolute for transitive classes. [3] A transitive set (or class) that is a model of a formal system of set theory is called a transitive model of the system (provided that the element relation of the model is the restriction of the true element relation to the universe of the model ...
A transitive relation is irreflexive if and only if it is asymmetric. [13] For example, "is ancestor of" is a transitive relation, while "is parent of" is not. Connected for all x, y ∈ X, if x ≠ y then xRy or yRx. For example, on the natural numbers, < is connected, while "is a divisor of " is not (e.g. neither 5R7 nor 7R5). Strongly connected
The action of G on X is called transitive if for any two points x, y ∈ X there exists a g ∈ G so that g ⋅ x = y. The action is simply transitive (or sharply transitive, or regular) if it is both transitive and free. This means that given x, y ∈ X the element g in the definition of transitivity is unique.
Hence the three defining properties of equivalence relations can be proved mutually independent by the following three examples: 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;
If M is a transitive model, then ω M is the standard ω. This implies that the natural numbers, integers, and rational numbers of the model are also the same as their standard counterparts. Each real number in a transitive model is a standard real number, although not all standard reals need be included in a particular transitive model.
For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by in the "Symmetric" column and in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle ...
The transitive closure of this relation is a different relation, namely "there is a sequence of direct flights that begins at city x and ends at city y". Every relation can be extended in a similar way to a transitive relation. An example of a non-transitive relation with a less meaningful transitive closure is "x is the day of the week after y".