In set theory, a branch of mathematics, a set A {\displaystyle A} is called transitive if either of the following equivalent conditions holds: whenever...

complement of a set (the class of all sets not in the given set). A set A {\displaystyle A\,} is called admissible if it is transitive and ⟨ A , ∈ ⟩ {\displaystyle...

alternating group is (n − 2)-transitive but not (n − 1)-transitive. The action of the general linear group of a vector space V on the set V ∖ {0} of non-zero vectors...

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...

defined as transitive sets of transitive sets. The induction situation in the first infinite ordinal ω {\displaystyle \omega } , the set of natural numbers...

In mathematical set theory, a transitive model is a model of set theory that is standard and transitive. Standard means that the membership relation is...

that found on the wiki page Transitive closure, i.e. "the transitive closure R+ of a homogeneous binary relation R on a set X is the smallest relation...

which is a subset of the power set of L α {\displaystyle L_{\alpha }} . Consequently, this is a tower of nested transitive sets. But L {\displaystyle L} itself...

In the mathematical field of graph theory, a transitive reduction of a directed graph D is another directed graph with the same vertices and as few edges...

Look up transitivity or transitive in Wiktionary, the free dictionary. Transitivity or transitive may refer to: Transitivity (grammar), a property regarding...

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,...

a set x: x is a (von Neumann) ordinal, x is a transitive set, and set membership is trichotomous on x, x is a transitive set totally ordered by set inclusion...

ordinals transitive 1.  A transitive relation 2.  The transitive closure of a set is the smallest transitive set containing it. 3.  A transitive set or class...

antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P = ( X , ≤ ) {\displaystyle P=(X,\leq )} consisting of a set X {\displaystyle...

V_{\alpha }} for some ordinal α {\displaystyle \alpha } . Any stage is a transitive set, hence every y ∈ x {\displaystyle y\in x} is already y ∈ V α {\displaystyle...

sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets...

principles. What follows concerns set theoretical concepts: The bounded notion of a transitive set of transitive sets is a good way to define ordinals...

relation. In this way, every finite partially ordered set can be represented as a DAG. The transitive reduction of a DAG is the graph with the fewest edges...

real number such as 0.75. An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfies all the...

In set theory, a supertransitive class is a transitive class which includes as a subset the power set of each of its elements. Formally, let A be a transitive...

subtle cardinal ≤ κ {\displaystyle \leq \kappa } if and only if every transitive set S {\displaystyle S} of cardinality κ {\displaystyle \kappa } contains...

where transitive ( x ) {\displaystyle {\text{transitive}}(x)} asserts that x {\displaystyle x} is transitive. Starting with the observation that set parameters...

a set of dice is intransitive if the binary relation – X rolls a higher number than Y more than half the time – on its elements is not transitive. More...

In set theory, the complement of a set A, often denoted by A c {\displaystyle A^{c}} (or A′), is the set of elements not in A. When all elements in the...

Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are...

is usually proved by forcing, whereby it is shown that every countable transitive model of ZFC (sometimes augmented with large cardinal axioms) can be expanded...

is a cardinal κ in a model of Zermelo–Fraenkel set theory with the property that for every transitive set M that includes κ and α < κ, there is a nontrivial...

mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable...

O r d {\displaystyle Ord} of all ordinals is a set. Then O r d {\displaystyle Ord} is a transitive set well-ordered by ∈ {\displaystyle \in } . So, by...

the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories...