Relational Algebra - SQL

It describes some operators, that together constitute an algebra that is not only relationally complete but also irreducibly so (very nearly- apart from RENAME, which can be expressed in terms of extension and projection, none of those operators can be discarded without sacrificing completeness). We can use these operators as a basis for testing SQL for relational completeness. If we can show that for every invocation of one of these operators there is an equivalent SQL expression, then we will have shown that SQL is relationally complete. By "equivalent" we mean an expression whose table operands are counterparts of the relation operands (ignoring the ordering that SQL imposes on the columns) and whose result is a table counterpart result, where a table is a counterpart of a relation if and only if it satisfies all of the following conditions:

• Every column has a name
• No two distinct columns have the same name
• No row appears more than once
• Null doesn't appear in place of a value anywhere in the table
• Every row consists entirely of its column values and doesn't somehow contain any additional data