Domain Restriction on Relation

1. Domain Restriction on Relation domain restriction operator , , restricts a relation to only those members whose domain is in a specified set. Let R: M x P be { (m1, p1), (m2, p2), (m3, p1), (m3, p3), (m4, p3) } and S = { m1,m3}, then : S R = { (m1, p1), (m3,p1), (m3, p3) } this can be an example of restricting to only those packages that contain module 1 and module 3. Note that S must be a subset of M in order for this to be sensible.

2. Domain Restriction • Given a set S and a relation R, domain restriction operation may also be expressed as id(S) ; R • S R = id(S) ; R • Consider the previous example of S and R in matrix form: 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 ; = 0 0 1 0 1 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 S R : M x P id (S) : M x M R: M x P Note that id(S) is a bit of a stretch here; it is really - - id(DOM R)\id(DOM R - S) ?? - - - - - - this is still not quite right ?!

3. Range Restriction on Relation range restriction operator , , restricts a relation to only those members whose range is in a specified set. Let R: M x P be { (m1, p1), (m2, p3), (m3, p2), (m3, p3), (m4, p3) } and S = { p1,p2}, then : R S = { (m1, p1), (m3,p2) } this can be an example of restricting to only those modules that are packaged in package 1 and package 2. This time, S needs to be a subset of P to make sense.

4. Range Restriction • Given a set S and a relation R, range restriction operation may also be expressed as R ; id(S) • R S = R ; id(S) • Consider the previous example of S and R in matrix form: 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 ; = 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 id (S) : P x P R S : M x P R: M x P Note that id(S) is a stretch; it is really - - - id(RAN R)\ id (RAN R- S)?

5. Additional Domain and Range restrictions • Domain restriction may be modified to restrict the relation, R, to only those members whose domain is NOT in the specified set S. • S R • Similarly, range restriction may be modified to restrict the relation, R, to only those members whose range is NOT in the specified set S. • R S

6. NOT restrictions on domain and range • Let R: A x B , S be a proper subset of A, and T be a proper subset of B. Then • S R = ( A \ S ) R • R T = R (B \ T )

7. Override Operator • The override operator over two relations R and S, where both relations are defined over the same sets A x B, is defined as follows: R S = (domS R) U S

8. Override Operator Example • Let R = { (J, m1), (T,m3), (K, m4), (S,m5) } and S = { (T,m2), (K,m3), (Q,m3) } defined over the same sets A x B where A = {J,T,K,S,Q} and B = { m1,m2,m3,m4,m5} • then R S = { (J,m1), (T,m2), (K,m3), (Q,m3), (S,m5) } • Note that the original relation R is “modified” with the members in S via: • Looking at the domain of S = T, K, Q • Consider only those members in R whose domain is NOT T,K, or Q • That leaves (J,m1) and (S,m5) in R. • Union of { (j,m1), (S,m5) } and S gives us the above result. • The override operator is very much like the file update capability that is often encountered in a software File System.

9. Relational IMAGE operator • Imageoperator applied to a relation, R, is the same as asking for the range of R. • If the Image operator is further restricted to a set S to be applied to the domain of R, then we have the Relational Image Operator, R( S ) • Relational Image Operator applied to a relation R as restricted by a set S returns the range of those members of R whose domain is in the set S. • Let R= {(a,2), (b,23), (c,11), (d,45)}, S= {b,c} • Then R (|S |) = { 23, 11 } Imagine the above R was a “table” of name and address. Then R( S ) would be the equivalent to a “query” look-up for the addresses of people specified in S.

10. Transitive Closure • Recall that earlier we defined an n-fold relation to be Rn. • Let R be a Homogeneous relation R: A x A , and A is some set. • R0 = id (A) • Rn = R ; R n-1 • Consider an example : A= {m1, m2, m3} and R is defined : AxA as R = {(m1,m2), (m1,m3), (m3, m2)}, then: • R0 = {(m1,m1), (m2,m2), (m3,m3)} • R1 = R; R0 = {(m1,m2), (m1,m3), (m3,m2)} • R2 = R ; R1 = { (m1,m2) } • R3 = R ; R2 = { } • Transitive Closure may be defined as: • Reflexive Transitive Closure of R, R* = R0 U R1 U R2 U - - - U Rn • Non-Reflexive Transitive Closure of R, R+ = R1 U R2 U - - - U Rn • Non-Reflexive Transitive Closure of R = Reflexive Transitive Closure of R \ R0 or R+ = R*\R0 Check the definition on page 132 – is it correct?

11. Non-Reflexive Transitive Closure • From the example on previous slide, • Non-Reflexive Transitive Closure = R1 U R2 • Since R2 is a subset of R1 in this case R1 becomes the non-reflexive transitive closure. • Depending on the interpretation one puts on the relation, different answers may be obtained. • If R, in this case, means module calling other modules, then the “max levels of calls” is the same as the largest n where Rn is non-empty.

12. A Theorem Involving Relation • Let: R1 be T1 x T2 and R2 be T2 x T3 • ├ (R1;R2)-1 = R2-1 ; R1-1 • (R1;R2)-1 means x (R1; R2)-1 y • x (R1; R2)-1 y implies that y (R1; R2) x • y (R1; R2) x means there is a z in T2 such that • y R1 z and z R2 x • y R1 z implies z R1-1 y and • z R2 x implies x R2-1 z • z R1-1 y “and” x R2-1 z is the same as x R2-1 z “and” z R1-1 y • x R2-1 z and z R1-1 y means R2-1; R1-1

13. Theorems involving Relations • See page 134 of your text. • You will not be held accountable of these theorems, but at least read them. • They will be given to you if ever needed.