Transcript Operations on Relations

Operations on Relations
•
•
R, complementary relation: aRb if and only if aRb
R-1, inverse relation: bR-1a if and only if aRb
(R-1)-1=R; Dom(R-1)=Ran(R) and
Dom(R)=Ran(R-1)
• MRS=MRMS; MRS=MRMS; MR-1=(MR)T; MR=MR
• R is symmetric if and only if R=R-1
Theorem 1. Suppose that R and S are relations from
A to B.
(a) if RS, then R-1S-1.
(b) if RS, then SR.
(c) (RS)-1=R-1S-1 and (RS)-1 =R-1S-1
(d) RS=RS and RS=RS
Operations on Relations
Theorem 2. Let R and S be relations on A.
(a) if R is reflexive, so is R-1
(b) if R and S are reflexive, then so are RS
and RS
(c) R is reflexive if and only if R is irreflexive
Theorem 3. Let R be a relation on a set A. Then
(a) R is symmetric if and only if R= R-1
(b) R is antisymmetric if and only if RR-1 (
is the equality relation on A)
(c) R is asymmetric if and only if RR-1=
Operations on Relations
Theorem 4. Let R and S be relations on A.
(a) If R is symmetric, so are R-1 and R.
(b) If R and S are symmetric, so are RS
and RS.
Theorem 5. Let R and S be relations on A.
(a) (RS)2R2S2
(b) If R and S are transitive, so is RS.
(c) If R and S are equivalence relations, so
is RS.
Closures & Composition
• the closure of R with respect to a property is the
smallest relation R1 on A that contains R and
possesses the property
– the reflexive closure of R is R
– the symmetric closure of R is RR-1
– the graph of the symmetric closure of R is
simply the digraph of R with all edges made
bidirectional

• the composition of R and S, written S R, is a relation
from A to C defined as: a(S R)c if and only if for some
b in B, aRb and bSc, where a is in A and c is in C (S
following R: first R, then S)
Theorem 6. Let R be a relation from A to B and let S be
 B to C. Then if A1 is any subset of A,
a relation from
we have (S R)(A1)=S(R(A1))
Composition
• let A={a1,…,an}, B={b1,…,bp}, C={c1,…,cm},
suppose that MR=[rij], Ms=[sij], and MSR=[tij], then
tij=1 if and only if (ai,cj)SR, which means that
for some k, (ai,bk)R and (bk,cj)S: MSR=MRMs
• RR=R2 and MR =MRMR
2
Theorem 7. Let A, B, C, and D be sets, R a relation
from A to B, S a relation from B to C, and T a
relation from C to D. Then
T(SR)=(TS)R
Theorem 8. Let A, B, and C be sets, R a relation
from A to B, S a relation from B to C. Then
(SR)-1=R-1S-1