The relation schema, Person($\underline{\text{pid}}$, $city$), describes the city of residence for every person uniquely identified by $pid$. The following relational algebra operators are available: selection, projection, cross product, and rename.
To find the list of cities where at least 3 persons reside, using the above operators, the minimum number of cross product operations that must be used is
A relation r(A, B) in a relational database has 1200 tuples. The attribute A has integer values ranging from 6 to 20, and the attribute B has integer values ranging from 1 to 20. Assume that the attributes A and B are independently distributed. The estimated number of tuples in the output of σ(A>10)∨(B=18)(r) is ______
![GATE CSE 2019 Database Management System - Relational Algebra Question 5 English 1](https://imagex.cdn.examgoal.net/7kejek273qbhg/0bc4d4ed-eae6-4499-ace0-2d38a56bc12e/56eb1d50-f7ae-11e9-a7bc-b7b08ce8d541/file-7kejek273qbhh-320w.jpg)
How many tuples will be returned by the following relational algebra query?
![GATE CSE 2019 Database Management System - Relational Algebra Question 5 English 2](https://imagex.cdn.examgoal.net/7kejek273tha4/c98a4c7f-9cc5-47e3-adee-025bd1fd0aaf/aed2c6d0-f7ae-11e9-a7bc-b7b08ce8d541/file-7kejek273tha5-320w.jpg)
$$Q:$$ $$\,\,\,\,\,\,\,\,\,$$ $$r$$ $$\,\,\,\,$$ $$\bowtie$$ $$\,\,\,\,$$ $$\left( {{\sigma _{b < 5}}\left( s \right)} \right)$$
Let $$LOJ$$ denote the natural left outer-join operation. Assume that $$r$$ and $$s$$ contain no null values.
Which one of the following queries is NOT equivalent to $$Q$$?