Left Outer Join:

• In left outer join all the tuples of left relation remain part of the output. The tuples that gave a matching tuple in the second relation do have the corresponding tuple from the second relation.
• However, for the tuples of the left relation, which do not have a matching record in the right tuple have Null values against the attributes of the right relation.

Project Operation (∏)

• Projects column(s) that satisfy given predicate.
• Notation: ∏A1, A2, An (r)
• Where a1, a2, an are attribute names of relation r.

• Duplicate rows are automatically eliminated, as relation is a set.

  For example:

  ∏subject, author (Books)

  Selects and projects columns named as subject and author from relation Books.

Rename operation ( ρ )

• Results of relational algebra are also relations but without any name.
• The rename operation allows us to rename the output relation. Rename operation is denoted with small greek letter rho ρ.
• Notation: ρ x (E)
• Where the result of expression E is saved with name of x. Outer Join:
• In outer join all the tuples of left and right relations are part of the output.
• It meansthat all those tuples of left relation which are not matched with right relation are left as Null.
• Similarly all those tuples of right relation which are not matched with leftrelation are left as null.

Division

• Not supported as a primitive operator, but useful for exressing queries like:
  Find sailors who have reserved all boats.
• Precondition: in A/B, the attributes in B must be included in the schema for A. Also, the result has attributes A-B.
• SALES(supId, prodId);
• PRODUCTS(prodId);
• Relations SALES and PRODUCTS must be built using projections.
• SALES/PRODUCTS: the ids of the suppliers supplying ALL products.