Definition Formula and Examples of Set Operations
Till now the students have dealt with mainly four basic operations of mathematics i.e. addition, subtraction, multiplication and division. These operations are mainly applied to two or more numbers to obtain a result that is a combination of these numbers. For example, when we apply the operation of addition on two numbers suppose 7 and 2, we get the number 9. Likewise, set operations are a group of operations that are applied on two or more sets that combines them and results in a single set. The set operations consist of three types of operations namely the union of sets (U), the intersection of sets (⋂) and the difference between sets (-). Let us understand all the set operations with suitable examples:
Union of Sets
Suppose A and B are sets consisting of elements. Let's say set A = {4, 5, 6, 2, 1} and set B = {7, 8, 9, 0}.
A⋃B is read as ‘A union B’, that means ‘union’ is denoted as ‘⋃’.
Therefore, A ⋃ B = {4, 5, 6, 2, 1} ⋃ {7, 8, 9, 0}.
= {4, 5, 6, 2, 1, 7, 8, 9, 0}
Hence, we see that the union of sets A and B consists of all the elements that were in set A and set B respectively.
Intersections of Sets
Suppose A and B are sets consisting of elements. Let's say set A = {8, 9, 5, 4, 6, 2} and set B = {5, 2, 3, 1, 9}.
A∩B is read as ‘A intersection B’, that means ‘intersection’ is denoted as ‘∩’.
Therefore, A⋂B = {8, 9, 5, 4, 6, 2}⋂{5, 2, 3, 1, 9}.
= {5, 2, 9}
Hence, when we apply intersection on two sets it gives us the elements that are common or are a part of both the sets.
Difference of Sets
Suppose A and B are two sets consisting of elements. Let’s say set A = {5, 6, 8, 9, 0} and B = { 9, 6, 0, 7, 3}
A - B is read as ‘A minus B’, that means the difference or minus is denoted as ‘-’.
Therefore, A - B = {5, 6, 8, 9, 0} - { 9, 6, 0, 7, 3}
= {5,8}
We can see that the difference of two sets A and B results in the set of elements that are a part of set A but are not in set B.
Similarly, B - A = { 9, 6, 0, 7, 3} - {5, 6, 8, 9, 0}
= {7, 3}
Here also, B - A results in the set of elements that are a part of set B but are not in set A.
Solved Examples
1. If A = {6, 9, 8, 1}, B = {1, 7, 5, 2, 9}, C = {7, 6, 0, 1} and D = {0, 1}. Find:-
A ∩ B
B ∩ C
A ∩ C
B ∩ D
A ∩ D
A ∩ (B U C)
A ∩ (B U D)
Answer - (a) A ∩ B = {6, 9, 8, 1} ∩ {1, 7, 5, 2, 9}
= {1, 9}
(b) B ∩ C = {1, 7, 5, 2, 9} ∩ {7, 6, 0, 1}
= {1, 7}
(c) A ∩ C = {6, 9, 8, 1}⋂ {7, 6, 0, 1}
= {1, 6}
(d) B ∩ D = {1, 7, 5, 2, 9} ⋂ {0, 1}
= {1}
(e) A ∩ D = {6, 9, 8, 1} ⋂ {0, 1}
= {1}
(f) A ∩ (B U C) = {6, 9, 8, 1} ⋂ ({1, 7, 5, 2, 9} U {7, 6, 0, 1})
= {6, 9, 8, 1} ∩ {1, 7, 5, 2, 9} U {6, 9, 8, 1} ⋂ {7, 6, 0, 1}
= {1, 9} U {1, 6}
= {1, 9, 6}
(g) A ∩ (B U D) = {6, 9, 8, 1} ∩ ({1, 7, 5, 2, 9} U {0, 1})
= {6, 9, 8, 1} ∩ {1, 7, 5, 2, 9} U {6, 9, 8, 1} ∩ {0, 1}
= {1,9} U {1}
= {1, 9}
FAQs on What Are Set Operations in Mathematics
1. What are set operations in mathematics?
Set operations are mathematical methods used to combine or compare sets, such as union, intersection, difference, and complement. These operations help describe relationships between groups of elements.
- Union (A ∪ B): Elements in A or B or both.
- Intersection (A ∩ B): Elements common to both A and B.
- Difference (A − B): Elements in A but not in B.
- Complement (A'): Elements not in A (within a universal set).
2. What is the union of two sets?
The union of two sets is the set containing all elements that are in either set or in both sets. It is written as A ∪ B.
- If A = {1, 2, 3}
- And B = {3, 4, 5}
- Then A ∪ B = {1, 2, 3, 4, 5}
3. What is the intersection of two sets?
The intersection of two sets is the set of elements that are common to both sets. It is written as A ∩ B.
- If A = {1, 2, 3}
- And B = {2, 3, 4}
- Then A ∩ B = {2, 3}
4. What is the difference between union and intersection of sets?
The difference between union and intersection is that union combines all elements, while intersection includes only common elements. In symbols:
- A ∪ B: All elements in A or B or both.
- A ∩ B: Only elements common to both A and B.
5. What is the complement of a set?
The complement of a set is the set of elements in the universal set that are not in the given set. It is written as A' or Ac.
- If U = {1, 2, 3, 4, 5}
- And A = {1, 2, 3}
- Then A' = {4, 5}
6. What is the set difference A − B?
The set difference A − B is the set of elements that are in A but not in B. It is also called the relative complement of B in A.
- If A = {1, 2, 3, 4}
- And B = {3, 4, 5}
- Then A − B = {1, 2}
7. What are the basic laws of set operations?
The basic laws of set operations describe how sets behave under union, intersection, and complement. Important laws include:
- Commutative law: A ∪ B = B ∪ A, A ∩ B = B ∩ A
- Associative law: (A ∪ B) ∪ C = A ∪ (B ∪ C)
- Distributive law: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
- Identity law: A ∪ ∅ = A, A ∩ U = A
- Complement law: A ∪ A' = U, A ∩ A' = ∅
8. How do you solve problems using set operations?
To solve problems using set operations, identify the sets, choose the required operation, and apply the correct formula. Follow these steps:
- Write down all given sets clearly.
- Determine whether the question asks for union, intersection, difference, or complement.
- Apply the operation symbol (∪, ∩, −, or ').
- List the resulting elements without repetition.
9. What is an example of set operations with three sets?
An example of set operations with three sets shows how union and intersection extend beyond two sets. Consider:
- A = {1, 2}
- B = {2, 3}
- C = {2, 4}
- A ∪ B ∪ C = {1, 2, 3, 4}
- A ∩ B ∩ C = {2}
10. Why are set operations important in mathematics?
Set operations are important because they form the foundation of set theory, probability, logic, and many areas of mathematics. They help in:
- Solving probability problems using unions and intersections.
- Understanding logical statements in Boolean algebra.
- Organizing and classifying data sets.
- Representing relationships using Venn diagrams.
