1. 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.
if RS, then R-1S-1.
if RS, then SR.
(RS)-1=R-1S-1 and (RS)-1 =R-1S-1
RS=RS and RS=RS

2. Operations on Relations
Theorem 2. Let R and S be relations on A.
if R is reflexive, so is R-1
if R and S are reflexive, then so are RS and RS
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=

3. Operations on Relations
Theorem 4. Let R and S be relations on A.
If R is symmetric, so are R-1 and R.
If R and S are symmetric, so are RS and RS.
Theorem 5. Let R and S be relations on A.
(RS)2R2S2
If R and S are transitive, so is RS.
If R and S are equivalence relations, so is RS.

4. 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 a relation from B to C. Then if A1 is any subset of A, we have (S R)(A1)=S(R(A1))

5. 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 M =MRMR
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