Introduction to Sets
A set is a well-defined collection of objects.
Examples
Set of vowels = {a, e, i, o, u}
Set of natural numbers less than 5 = {1, 2, 3, 4}
The objects of a set are called elements or members.
Representation of Sets
1. Roster (Listing) Method
All elements are listed inside curly brackets.
Example:
F = {Chhiring, Dorje, Hari, Maya, Aasha}
2. Set-Builder Method
Elements are described using a rule.
Example:
A = {x : x is a positive even number less than 10}
Types of Sets
Universal Set
The set that contains all elements under discussion is called the universal set.
It is denoted by U.
Subset
If every element of A is also an element of B, then A is a subset of B.
Written as A ⊂ B
Proper Subset
If A ⊂ B and A ≠ B, then A is a proper subset of B.
Empty Set
A set with no elements.
Denoted by ∅.
Equal Sets
Two sets are equal if they contain the same elements.
Venn Diagram
.
Venn diagram two intersecting sets with universal set
Set Operations
Union of Sets ( ∪ )
The union of sets A and B contains all elements of A or B or both.
Symbol: A ∪ B
Example
A = {1, 2, 3}
B = {3, 4, 5}
A ∪ B = {1, 2, 3, 4, 5}
Set-builder form:
A ∪ B = {x : x ∈ A or x ∈ B}
Intersection of Sets ( ∩ )
The intersection contains only common elements of both sets.
Symbol: A ∩ B
Example
A = {1, 2, 3, 4}
B = {3, 4, 5}
A ∩ B = {3, 4}
Set-builder form:
A ∩ B = {x : x ∈ A and x ∈ B}
Difference of Sets ( − )
A − B contains elements in A but not in B.
Example
A = {1, 2, 3, 4}
B = {3, 4, 5}
A − B = {1, 2}
Similarly,
B − A = {5}
Symmetric Difference
Elements which are in A or B but not in both.
Symbol: A Δ B
A Δ B = (A − B) ∪ (B − A)
Complement of a Set
The complement of set A contains elements in U but not in A.
Symbol: A̅ or A′
Formula:
A′ = U − A
Example
U = {1,2,3,4,5}
A = {1,3,5}
A′ = {2,4}
Important Identities
A ∪ U = U
A ∩ U = A
A ∪ A′ = U
A ∩ A′ = ∅
Cardinality of Sets
The number of elements in a set is called its cardinality.
Notation: n(A)
Example
A = {a, b, c}
n(A) = 3
Cardinality Formulae
Disjoint Sets
If A and B have no common elements:
n(A ∪ B) = n(A) + n(B)
Intersecting Sets
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Only A
n(A only) = n(A) − n(A ∩ B)
Only B
n(B only) = n(B) − n(A ∩ B)
Solved Examples (Important)
Question 1
If A = {1, 2, 3} and B = {3, 4, 5}, find
(i) A ∪ B
(ii) A ∩ B
Solution
Given,
A = {1, 2, 3}
B = {3, 4, 5}
(i) Union (A ∪ B) means all elements of A and B without repetition.
A ∪ B = {1, 2, 3, 4, 5}
(ii) Intersection (A ∩ B) means common elements.
Common element in A and B is 3
A ∩ B = {3}
Question 2
If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and
A = {2, 4, 6, 8}, find A′ (complement of A).
Solution
Complement of A means elements in U but not in A.
U = {1,2,3,4,5,6,7,8,9,10}
A = {2,4,6,8}
Removing elements of A from U,
A′ = {1, 3, 5, 7, 9, 10}
Question 3
If n(A) = 20, n(B) = 15 and n(A ∩ B) = 5, find n(A ∪ B).
Solution
Formula for intersecting sets:
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Substitute values:
n(A ∪ B) = 20 + 15 − 5
n(A ∪ B) = 30
Question 4
In a class of 50 students,
30 like Mathematics,
25 like Science,
10 like both.
Find how many like at least one subject.
Solution
Let,
n(M) = 30
n(S) = 25
n(M ∩ S) = 10
Students who like at least one subject = n(M ∪ S)
Using formula:
n(M ∪ S) = n(M) + n(S) − n(M ∩ S)
n(M ∪ S) = 30 + 25 − 10
n(M ∪ S) = 45
Question 5
How many subsets can be formed from a set having 4 elements?
Solution
Formula:
Number of subsets = 2ⁿ
Here, n = 4
Number of subsets = 2⁴ = 16
Question 6
If A ⊂ B, n(A) = 12 and n(B) = 20, find n(A ∩ B).
Solution
If A is a subset of B, then all elements of A are common with B.
So,
A ∩ B = A
Therefore,
n(A ∩ B) = n(A) = 12
Question 7
In a survey of 100 people,
60 like tea,
50 like coffee,
20 like both.
Find how many like neither.
Solution
Let,
n(T) = 60
n(C) = 50
n(T ∩ C) = 20
n(U) = 100
First find n(T ∪ C):
n(T ∪ C) = 60 + 50 − 20
n(T ∪ C) = 90
People who like neither:
= n(U) − n(T ∪ C)
= 100 − 90
= 10
Question 8
If U = {1 to 20},
A = set of prime numbers,
B = set of multiples of 3,
find A ∩ B.
Solution
Prime numbers between 1 and 20:
A = {2, 3, 5, 7, 11, 13, 17, 19}
Multiples of 3 less than 20:
B = {3, 6, 9, 12, 15, 18}
Common element in A and B is 3.
So,
A ∩ B = {3}
Question 9
If n(U) = 200, n(A) = 120, n(B) = 90 and n(A ∩ B) = 40, find number of elements in A only.
Solution
Formula:
A only = n(A) − n(A ∩ B)
Substitute values:
A only = 120 − 40
A only = 80
Question 10
Verify De Morgan’s Law:
(A ∪ B)′ = A′ ∩ B′
Solution
According to De Morgan’s Law,
Complement of union equals intersection of complements.
So,
(A ∪ B)′ = A′ ∩ B′
Hence, verified.
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