1. Algebraic Laws

2. 16.2.1 Commutative and Associative Laws Commutativity for Sets and Bags (Ch5): R x S = S x R (Proof) R  S = S  R (ch5 e) R U S = S U R(ch5) R ∩ S = S ∩ R (ch5) Associativity Sets and Bags: : (R x S) x T = R x (S x T)‏ (R  S)  T = R  (S  T)‏ (R U S) U T = R U (S U T)‏ (ch5) (R ∩ S) ∩ T = R ∩ (S ∩ T)‏ (ch5)

3. 16.2.2 Laws Involving Selection For a union, the selection must be pushed to both arguments. σ c (R U S) = σ c (R) U σ c (S) For a difference, the selection must be pushed to first argument and optionally to second. σ c (R - S) = σ c (R) – S σ c (R - S) = σ c (R) - σ c (S) It is only required that the selection must be pushed to one or both argument. –σ c (R x S) = σ c (R) x S –σ c (R  S) = σ (R)  S –σ c (R  D S) = σ (R)  D S –σ c ( R ∩ S) = σ c (R) ∩ S

4. 16.2.2 Laws Involving Selection it is only required that the selection must be pushed to one or both argument. –σ c (R x S) = R x σ c (S) –σ c (R  S) = σ c (R)  σ c (S)

5. 16.2.4. Laws involving Projection Consider term π E  x –E : attribute, or expression involving attributes and constants. –All attributes in E are input attributes of projection and x is output attribute Simple projection: if a projection consists of only attributes. –Example: π a,b,c (R) is simple. a,b,c are input and output attributes. Projection can be introduced anywhere in expression tree as long as it only eliminates attributes that are never used.

6. 16.2.4. Laws involving Projection (cont….) π L (R S) = π L ( π M (R) π N (S)) ; M and N are all attributes of R and S that are either join (in schema of both R and S) or input attributes of L π L (R c S) = π L ( π M (R) c π N (S)) ; M and N are all attributes of R and S that are either join(mentioned in condition of C ) or input attributes of L π L (R x S) = π L ( π M (R) x π N (S)) ; M and N are all attributes of R and S that are input attributes of L Projections cannot be pushed below set unions or either of set or bag versions of intersection or difference at all.