Solution

The correct answer is R1 ⋂ R2 and R1 - (R1 - R2)

Key Points 

Intersection, Union and Difference operations are basic set operations that are used in many areas of mathematics and computer science, including databases and programming. For the explanation, let's think of sets (or relations, in database terms) as collections of distinct items (or tuples).

• Intersection (⋂): The intersection of two sets, R1 and R2, denoted as R1 ⋂ R2, is a set that contains all the tuples that are in both R1 and R2.
  • For instance, if R1 contains the elements {a, b, c} and R2 contains the elements {b, c, d}, the intersection of R1 and R2 would be {b, c} because these are the elements common to both sets.
• Union (⋃): The union of two sets, R1 and R2, denoted as R1 ⋃ R2, is a set that contains all the tuples that are in R1 or R2 or in both.
  • Continuing with the previous example, the union of sets R1 {a, b, c} and R2 {b, c, d} would be {a, b, c, d}. This includes all unique elements from both sets.
• Difference (-): The difference operation between two sets, R1 and R2, denoted as R1 - R2, results in a set that contains all the tuples that are present in R1 but not in R2.
  • So, if we take the difference between R1 {a, b, c} and R2 {b, c, d}, we would have {a}, which is the element present in R1 but not in R2.
  • In the context of relational databases, these operations act on tables or relations where tuples are equivalent to set elements. For instance, for the "difference" operation, the resulting relation includes all rows that are in the first relation (R1) and not in the second (R2).

Let's break it down:

• Operation 1: Intersection (R1 ⋂ R2)
  • The intersection of two sets is a set containing all the elements that are common to both sets.
  • So, in the case of R1 and R2, the intersection operation ⋂ will look at both relations (or sets of tuples), and return a new relation that contains only the tuples that appear in both R1 and R2. In this case, the tuple containing ("b1", 5) is the only one that appears in both relations, hence it is the result of the intersection.
• Operation 3: R1 - (R1 - R2)
  • "R1 - R2" stands for set difference, which is a set of elements that exists in one set but not the other. So, "(R1 - R2)" gives us all tuples that exist in R1 but not in R2.
  • But the operation we are looking at is "R1 - (R1 - R2)". This operation will give us all tuples from R1 that also exist in R2, which in terms of set theory, is the exact definition of intersection.
• So "R1 - (R1 - R2)" is essentially a long approach to defining an Intersection operation which is simply written as R1 ⋂ R2.

So, with the given data, both options 1) and 3) will provide us the common tuples between R1 and R2, which in this case is just ("b1", 5).