Properties of Relations

Report
Properties of Relations
• In many applications to computer science and applied
mathematics, we deal with relations on a set A rather
than relations from A to B
• A relation R on a set A is reflexive if (a,a)R for all
aA; it’s irreflexive if (a,a)R for all aA
• Let ={(a,a)| aA},  is the relation of equality on the
set A, and  is reflexive
• The matrix of a reflexive relation must have all 1’s on
its main diagonal, while irreflexive must have all 0’s on
its main diagonal.
• A reflexive relation has a cycle of length 1 at every
vertex, while irreflexive has no cycles of length 1.
• R is reflexive if and only if R, and R is irreflexive if
and only if R=
• If R is reflexive on a set A, then Dom(R)=Ran(R)=A.
Symmetric, Asymmetric and
Antisymmetric Relations
• A relation R on a set A is symmetric if whenever
aRb, then bRa
– R is not symmetric if there are some a and bA with
aRb, but bRa
• A relation R on a set A is asymmetric if
whenever aRb, then bRa
– R is not asymmetric if there are some a and bA
with both aRb and bRa
• A relation R on a set A is antisymmetric if
whenever aRb and bRa, then a=b;
– The contrapositive of this definition: R is
antisymmetric if whenever ab, then aRb or bRa
– R is not antisymmetric if there are some a and bA,
ab, and both aRb and bRa
Symmetric, Asymmetric and
Antisymmetric Relations
• MR =[mij] of a symmetric relation satisfies the
property that if mij=1, then mji=1 and if mij=0,
then mji=0; it’s a symmetric matrix
– mii can be either 0 or 1 for all i
• MR =[mij] of an asymmetric relation satisfies
the property that if mij=1, then mji=0 and mii=0
for all i
– If mij=0, then mji can be either 0 or 1
• MR =[mij] of an antisymmetric relation satisfies
the property that if ij, then mij=0, or mji=0
– mii can be either 0 or 1 for all i
Symmetric, Asymmetric and
Antisymmetric Relations
• If relation R is asymmetric, then the digraph
of R cannot simultaneously have an edge
from vertex i to j and an edge from j to i; and
there can be no cycles of length 1: all edges
are one-way
• If relation R is antisymmetric, then for
different vertices i and j there cannot not be
an edge from i to j and an edge from j to i;
when i=j, no condition is imposed, thus there
may be cycles of length 1: still all edges are
one-way
• If relation R is symmetric, then whenever
there is an edge from vertex i to j, then there
is an edge from vertex j to i.
Transitive Relations
•
•
•
•
A relation R on a set A is transitive if whenever aRb
and bRc, then aRc.
A relation R on A is not transitive if there exists a, b,
and c in A so that aRb and bRc, but aRc.
– If such a, b, and c do not exist, then R is transitive
R is transitive if and only if its matrix MR =[mij] has the
property if mij=1 and mjk=1, then mik=1.
If MRMR has a 1 in any position, then MR must have a
1 in the same position, i.e. if MRMR = MR, then R is
transitive
– The converse is not true, i.e. if MRMR  MR, then
R may be transitive or may not be transitive
Transitive Relations
Theorem 1. A relation R is transitive if and only if it satisfies
the following property: if there is a path of length
greater than 1 from vertex a to vertex b, there is a path
of length 1 from a to b (that is, a is related to b).
Algebraically stated, R is transitive if and only if RnR
for all n1.
Theorem 2. Let R be a relation on a set A, then
(a) Reflexive of R means that aR(a) for all a in A.
(b) Symmetry of R means that aR(b) if and only if
bR(a).
(c) Transitivity of R means that if bR(a) and cR(b), then
cR(a) .
Equivalence Relations and Partitions
• A relation R on a set A is called an equivalence relation if it
is reflexive, symmetric and transitive.
Theorem 1. Let P be a partition of a set A. Recall that the sets
in P are called the blocks of P. Define the relation R on A as
follows:
aRb if only if a and b are members of the same block. Then
R is an equivalence relation on A.
• R will be called the equivalence relation determined by P
– each element in a block is related to every element in
the same block and only to these elements.
Lemma 1. Let R be an equivalence relation on A, and let aA
and bA, then aRb if and only if R(a)=R(b).
Equivalence Relations and Partitions
Theorem 2. Let R be an equivalence relation on A,
and let P be the collection of all distinct relative
sets R(a) for a in A. Then P is a partition of A and
R is the equivalence relation determined by P.
• If R is an equivalence relation on A, then the sets
R(a) are called equivalence classes of R.
• The partition P constructed in Theorem 2 consists
of all equivalence classes of R, denoted by A/R
• P is the quotient set of A, that is constructed from
and determines R.
Equivalence Relations and
Partitions
A general procedure for determining partitions A/R for a
set A which is finite or countable:
step1: choose any element a of A and compute the
equivalence class R(a)
step2: if R(a)A, choose an element b, not included in
R(a), and compute the equivalence class R(b)
step3: if A is not the union of previously computed
equivalence classes, then choose an element x of A
that is not in any of those equivalence classes and
compute R(x)
step4: repeat step3 until all element of A are included in
the computed equivalence classes. If A is countable,
this process could continue indefinitely. In that case,
continue until a pattern emerges that allows to
describe or give a formula for all equivalence classes.

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