# Transitivity

A relation R ⊆ X2 is transitive iff, whenever Rxy and Ryz, then also Rxz.

**Transitive** relation R on set A is called Transitive if xRy and yRz implies xRz,∀x,y,z∈A. Example: The relation R={(1,2),(2,3),(1,3)} on set A={1,2,3} is transitive.

## Transitive relations

A binary relation `R` over a set `X` is transitive if whenever an element `a` is related to an element `b` and `b` is related to an element `c` then `a` is also related to `c`.

Transitivity (or transitiveness) is a key property of both _partial order_ relations and _equivalence_ relations.

Formal definition: `∀ a,b,c ∈ X: a ⥽ b ∧ b ⥽ c ⇒ a ⥽ c`

## transitive

For example, "is ancestor of" is transitive, while "is parent of" is not. A transitive relation is irreflexive if and only if it is asymmetric.

## PM

(terms from Principia Mathematica that are no longer widely used or whose meaning has changed)

A relation R is called **compact** if whenever xRz there is a y with xRy and yRz

connexity, connected: A relation R is called **connected** if for any 2 distinct members x, y either xRy or yRx. (now called connex?)

definiendum The symbol being defined

definiens The meaning of something being defined

first-order A first-order proposition is allowed to have quantification over individuals but not over things of higher type.

second-order A second order function is one that may have first-order arguments

molecular proposition A proposition built from two or more atomic propositions using "or" and "not"; in other words an elementary proposition that is not atomic.

predicative A century of scholarly discussion has not reached a definite consensus on exactly what this means, and Principia Mathematica gives several different explanations of it that are not easy to reconcile. ...a predicative function is one with no apparent (bound) variables, in other words a matrix.

referent The term x in xRy

relatum The term y in xRy

reflexive infinite in the sense that the class is in one-to-one correspondence with a proper subset of itself (\*124)

relation A propositional function of some variables (usually two). This is similar to the current meaning of "relation".

serial relation A total order on a class
