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transitiveness

Transitivity, or the transitive property, is a characteristic of a binary relation R on a set X. R is transitive if for all elements a, b, and c in X, whenever aRb and bRc hold, then aRc must also hold.

Common examples illustrate transitivity. Equality is transitive: if a = b and b = c, then a = c.

Not all relations are transitive. The parent relation is not transitive: if x is a parent of

Related concepts and constructions include the transitive closure of a relation, the smallest transitive relation containing

The
subset
relation
is
transitive:
if
A
⊆
B
and
B
⊆
C,
then
A
⊆
C.
In
number
theory,
if
b
divides
a
and
c
divides
b,
then
c
divides
a.
In
family
terms,
if
x
is
an
ancestor
of
y
and
y
is
an
ancestor
of
z,
then
x
is
an
ancestor
of
z.
y
and
y
is
a
parent
of
z,
x
is
not
necessarily
a
parent
of
z
(x
is
a
grandparent).
The
relation
“is
friends
with”
is
generally
not
transitive
either,
since
friends
of
friends
are
not
guaranteed
to
be
friends.
the
original
one;
this
is
important
in
computing
reachability
in
graphs.
In
algebra
and
logic,
transitivity
interacts
with
other
properties
such
as
reflexivity,
symmetry,
and
antisymmetry.
Equivalence
relations
are
reflexive,
symmetric,
and
transitive;
partial
orders
are
reflexive,
antisymmetric,
and
transitive,
illustrating
how
transitivity
fits
into
broader
classifications
of
relations.