Editing Intransitivity (section)

== Intransitivity ==

A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C.  A relation is {{em|intransitive}} if it is not transitive.  Assuming the relation is named <math>R</math>, it is intransitive if:

<math display=block>\lnot\left(\forall a, b, c: a R b \land b R c \implies a R c\right).</math>

This statement is equivalent to
<math display=block>\exists a,b,c : a R b \land b R c \land \lnot(a R c).</math>

For example, the inequality relation, <math>\neq</math>, is intransitive.  This can be demonstrated by replacing <math>R</math> with <math>\neq</math> and choosing <math>a=1</math>, <math>b=2</math>, and <math>c=1</math>.  We have <math>1\neq 2</math> and <math>2\neq 1</math> and it is not true that <math>1\neq 1</math>.

Notice that, for a relation to be intransitive, the transitivity condition just has to be not true at some <math>a</math>, <math>b</math>, and <math>c</math>.  It can still hold for others.  For example, it holds when <math>a=1</math>, <math>b=2</math>, and <math>c=3</math>, then <math>1\neq 2</math> and <math>2\neq 3</math> and it is true that <math>1\neq 3</math>.

For a more complicated example of intransitivity, consider the relation ''R'' on the integers such that ''a R b'' if and only if ''a'' is a multiple of ''b'' or a divisor of ''b''. This relation is intransitive since, for example, 2 ''R'' 6 (2 is a divisor of 6) and 6 ''R'' 3 (6 is a multiple of 3), but 2 is neither a multiple nor a divisor of 3. This does not imply that the relation is {{em|antitransitive}} (see below); for example, 2 ''R'' 6, 6 ''R'' 12, and 2 ''R'' 12 as well.

An example in biology comes from the [[food chain]].  Wolves feed on deer, and deer feed on grass, but wolves do not feed on grass.<ref>Wolves ''do'' in fact eat grass – see {{cite book|title=Wild Health: Lessons in Natural Wellness from the Animal Kingdom|first1=Cindy|last1=Engel|year=2003|edition=paperback|publisher=Houghton Mifflin|isbn=0-618-34068-8|page=141}}.</ref> Thus, the {{em|feed on}} relation among life forms is intransitive, in this sense.