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Explain the following relational algebra operations with proper examples

The terms are

i. Left outer join

ii. Division

iii. Rename

iv. Outer join

v. Project

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### 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 expressing 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.