# Transitivity Transitivity * transitive * intransitive * antitransitive * quasitransitive The notation `a𝓡b` is the infix notation for `(a, b) ∈ 𝓡` ## Transitive A homogeneous relation `R` on the set `X` is a transitive relation if `∀abc ∈ X. (a𝓡b ∧ b𝓡c) -> a𝓡c` ## Intransitive A homogeneous relation `R` on the set `X` is a intransitive (non-transitive) relation if `¬(∀abc ∈ X. (a𝓡b ∧ b𝓡c) -> a𝓡c)` <=> `∃abc ∈ X. a𝓡b ∧ b𝓡c ∧ ¬a𝓡c` For instance, wolves feed on deer, and deer feed on grass, but wolves do not feed on grass. Thus, this feed-on relation is intransitive, in this sense. The feed-on relation is intransitive, but it still contains some transitivity, e.g. humans feed on rabbits, rabbits feed on carrots, and humans also feed on carrots. ## Antitransitive Often the term intransitive is used to refer to both non-transitivity and the stronger property of *antitransitivity*. > A relation is *antitransitive* if this never occurs at all: `∀abc ∈ X. (a𝓡b ∧ b𝓡c) -> ¬a𝓡c` An example of an antitransitive relation: the `has-defeated` relation in knockout tournaments. If player A defeated player B and player B defeated player C, A could never have played C, and therefore, A has not defeated C. By transposition, each of the following formulas is equivalent to antitransitivity of R: * `∀abc ∈ X. (a𝓡b ∧ b𝓡c) -> ¬a𝓡c` * `∀abc ∈ X. (a𝓡b ∧ a𝓡c) -> ¬b𝓡c` * `∀abc ∈ X. (a𝓡c ∧ b𝓡c) -> ¬a𝓡b`