What Are the Key Rules for Intersection and Difference of Sets?
Set Operations
In our everyday life, we deal with collections of objects (person, numbers or any other thing).
For example, consider the following collections:
i) Collection of all the students in your class.
ii) Collection of all the teachers at your school.
iii) Collection of all counting numbers that are less than 10 i.e., of numbers 1, 2, 3, 4, 5, 6, 7, 8, 9.
iv) Collection of all even natural numbers less than 15 i.e., of numbers 2, 4, 6, 8, 10, 12, 14.
v) Collection of the first five natural numbers divisible by 5 i.e., of the numbers 5,10, 15, 20, 25.
vi) Collection of all vowels in the English alphabets i.e., of the letters a, e, i, o, u.
vii) Collection of all the days in a week.
viii) Collection of all the books in your bag.
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Note: All the above collections is a well-defined collection of objects.
A ‘well-defined collection of objects’ means that if we are provided with a collection and an object, then it would be possible to assert without any doubt that if an object belongs to the collection or not.
What is the Data Set?
A set is a defined collection of objects
The objects that belong to the set are called its members or elements. Each of the above collections is a set.
Now consider the following collections:
i) Collection of all the intelligent students in your class.
We cannot call it a well-defined collection because people may differ on whether a student of your class is intelligent or not.
ii) Collection of all the competent teachers of your school.
We cannot call it a well-defined collection because people may differ on whether a teacher of your school is competent or not.
iii) Collection of four days of a week.
We cannot call it a well-defined collection because it is not known which four days of a week are to be included in the collection.
Any of the above collections are not a set.
Notation
The sets are usually denoted by capital letters A, B, C, and so on… The members of a set are denoted by small letters x, y, z, and so on.
If x is a member of the set A, we write x ∈ A (read as ‘x belongs to A’) and if x is not a member of the set A, we write x ∉ A (read as ‘x does not belong to A’)
If x and y are the members of the set A, we write x, y ∈ A.
Representation of A Set
A set can be represented by the following method:
Description method
Roster method or tabular form
Rule method or set builder form.
Types of Sets
There are Four Types of Sets
Union of Sets
The union of two sets A and B is the set consisting of all the elements which belong to either A or B or both. We write it as A U B.
Example, if A = { a, e, i, o, u } and B = { a, b, c, d, e } the,
A U B = { a, e, i, o, u, b, c, d }
Intersection of Sets
The intersection of two sets A and B is the set consisting of all elements which belong to both A and B. we write it as A ∩ B.
Example Of Intersection Of Sets
i) If A = {a, e, i, o, u} and B = {a, b, c, d, e} then,
A ∩ B= {a,e}
ii) If A = {the colours of the rainbow} = {Violet, Indigo, Blue, Green, Yellow, Orange, Red}
And B = {Black, Red, Blue, White} the
A ∩ B = {Red, Blue}
Difference of Sets
The difference of two sets A and B is a set with no elements in common.
For example,
i) A = {1, 3, 5, 7, 9} and B = {0, 2, 4, 6, 8, 10}
There is a difference of two sets A and B as there are no common elements between them.
Properties of Set Operations
Union of Sets
Properties of (A U B) are:
The commutative law holds true as (A U B) = (B U A)
The associative property too holds true as (A U B) U {C} = {A} U (B U C)
Intersection of Sets
Properties of – A ∩ B are:
Commutative law = (A ∩ B) ∩ C = A ∩ (B ∩ C)
Associative law = A ∩ B = B ∩ A
Distributive law = A ∩ (B U C) = (A ∩ B) U (A ∩ C)
φ ∩ A = φ
U ∩ A = A
FAQs on Operations on Sets: Intersection and Difference Explained
1. What is the intersection of two sets as per the CBSE Class 11 syllabus for 2025-26?
The intersection of two sets, say A and B, is the set containing all elements that are common to both A and B. It is denoted by the symbol A ∩ B. In set-builder notation, it is written as A ∩ B = {x : x ∈ A and x ∈ B}. For example, if A = {1, 2, 3} and B = {2, 3, 4}, then their intersection A ∩ B = {2, 3}.
2. What does the difference of two sets mean with an example?
The difference of two sets, A and B, denoted as A - B, is the set of all elements that are in set A but not in set B. It is crucial to note the order of the sets matters. For instance, if A = {a, b, c, d} and B = {c, d, e, f}, then A - B = {a, b}. Conversely, the difference B - A would be {e, f}.
3. How are the intersection and difference of sets represented using a Venn diagram?
Venn diagrams provide a clear visual representation of set operations:
Intersection (A ∩ B): This is represented by the overlapping area of the circles for set A and set B. This region contains all the elements that are present in both sets.
Difference (A - B): This is represented by the part of the circle for set A that does not overlap with the circle for set B. It shows the elements that are exclusively in A.
4. If A = {2, 4, 6, 8} and B = {6, 8, 10, 12}, what are A ∩ B and A - B?
To find the intersection and difference for the given sets:
Intersection (A ∩ B): We look for elements that are in both A and B. The common elements are 6 and 8. Therefore, A ∩ B = {6, 8}.
Difference (A - B): We look for elements that are in A but not in B. The elements 2 and 4 are in A but not in B. Therefore, A - B = {2, 4}.
5. What is the fundamental difference between the operations A - B and B - A? Can they ever be equal?
The operations A - B and B - A are fundamentally different because the difference of sets is not commutative.
- A - B includes elements that are only in set A.
- B - A includes elements that are only in set B.
They can only be equal in one specific case: when A = B. In this situation, both A - B and B - A result in the empty set (∅), making them equal.
6. What happens if the intersection of two non-empty sets is an empty set (A ∩ B = ∅)?
If the intersection of two non-empty sets A and B is the empty set (∅), it means that there are no common elements between them. Such sets are called disjoint sets. For example, the set of even numbers E = {2, 4, 6, ...} and the set of odd numbers O = {1, 3, 5, ...} are disjoint sets because their intersection E ∩ O is ∅.
7. How are the concepts of set intersection and difference applied in practical scenarios?
Set operations have many real-world applications. For instance:
Intersection: In database management, finding records that satisfy multiple conditions (e.g., customers who are 'members' AND live in a specific 'city') is an application of intersection.
Difference: In marketing analytics, a company might want to find customers who purchased Product A but NOT Product B (A - B) to send them targeted promotions for Product B.
8. How is the symmetric difference of two sets related to their difference and intersection?
The symmetric difference of two sets A and B, denoted A Δ B, is the set of elements which are in either of the sets, but not in their intersection. It can be understood through the operations of difference and union. There are two common ways to define it:
1. As the union of the two differences: A Δ B = (A - B) ∪ (B - A).
2. Using union and intersection: A Δ B = (A ∪ B) - (A ∩ B).
Essentially, it captures all elements that are not common to both sets.
