Finite Set

1.0What is a Finite Set?

A finite set is a set that contains a finite number of elements. In simpler terms, you can count all the elements in the set, and the counting process will eventually come to an end. The number of elements in a finite set is a whole number (non-negative integer).

Formally, a set A is called a finite set if it is either an empty set or can be put into one-to-one correspondence with the set of the first n natural numbers, {1,2,3,...,n} for some non-negative integer n.

The most important characteristic of a finite set is that its elements are countable.

2.0Examples of Finite Sets

Here are some clear examples of finite sets:

• The set of all days in a week: A={Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
• The set of vowels in the English alphabet: V={a,e,i,o,u}
• The set of positive divisors of 12: D={1,2,3,4,6,12}
• The set of students in your school.
• The set of solutions to the equation $x^2-4=0$ which is S={−2,2}.

All these sets have a limited, countable number of elements.

3.0Cardinality of a Finite Set

The cardinality of a finite set is the number of elements in that set. It is also known as the cardinal number of the set. The cardinality of a set A is denoted by n(A), ∣A∣, or #(A).

For the set of vowels V={a,e,i,o,u}, the cardinality is n(V)=5.

The cardinality of the set of days in a week, A, is n(A)=7.

Solving finite-set problems requires an understanding of cardinality, particularly when applying the Principle of Inclusion-Exclusion..

4.0Types of Finite Sets

1. Empty Set: A set with no elements is referred to as an empty set, null set, or void set. Its cardinality is zero. It's represented by ∅ or {}. Because it contains zero elements, the empty set is a finite set.

• Example: The set of all odd numbers divisible by 2 is an empty set.

2. Singleton Set: A singleton set is a set that contains exactly one element. Its cardinality is one.

• Example: The set of even prime numbers, P={2}, is a singleton set.

3. Equal Sets: Two finite sets, say A and B, are called equal sets if they have the same elements. The order of elements does not matter. If A=B, then n(A)=n(B).

• Example: If A={1,2,3} and B={3,1,2}, then A=B.

4. Equivalent Sets: Two finite sets, which will be denoted A and B, are equivalent (equivalently, of equal cardinality) if they each have the same number of elements. Importantly, the particular elements in the sets do not have to be the same! To think about this equivalence, consider that the sets can be connected by a one-to-one correspondence of their elements.

• If n(A)=n(B), then A and B are equivalent sets. This is denoted by A∼B.
• Example: If A={1,2,3} and B={a,b,c}, then n(A)=3 and n(B)=3. Therefore, A∼B.

Note: All equal sets are equivalent, but not all equivalent sets are equal.

5.0Operations on Finite Sets

When dealing with finite sets, we can perform several operations to create new sets.

1. Union of Sets: The union of two sets, A and B, denoted by A∪B, is the set of all elements that are in A, or in B, or in both.

• A∪B={x∣x∈A or x∈B}

2. Intersection of Sets: The intersection of two sets, A and B, denoted by A∩B, is the set of all elements that are common to both A and B.

• A∩B={x∣x∈A and x∈B}
• If A∩B=∅, the sets are called disjoint sets.

3. Difference of Sets: The difference of two sets, A and B, denoted by A−B, is the set of all elements that are in A but not in B.

• A−B={x∣x∈A and x∈/B}

4. Symmetric Difference: The symmetric difference of two sets, A and B, denoted by AΔB, is the set of elements that are in either A or B, but not in their intersection.

• AΔB=(A−B)∪(B−A) or AΔB=(A∪B)−(A∩B)

5. Complement of a Set: The complement of a set A, denoted by A^c or A′, is the set of all elements in the universal set U that are not in A.

• $A^c=U-A=$ { $x\mid x\in Uandx\in /A$ }

6. Cartesian Product: The Cartesian product of two sets, A and B, denoted by A×B, is the set of all ordered pairs (a,b) where a∈A and b∈B.

• If A={1,2} and B={3,4}, then A×B={(1,3),(1,4),(2,3),(2,4)}. The cardinality of the Cartesian product is given by n(A×B)=n(A)×n(B).

6.0Properties of Finite Sets

Finite sets have several important properties that are crucial for JEE-level problems.

1. Commutative Laws:

• A∪B=B∪A
• A∩B=B∩A

2. Associative Laws:

• (A∪B)∪C=A∪(B∪C)
• (A∩B)∩C=A∩(B∩C)

3. Distributive Laws:

• A∪(B∩C)=(A∪B)∩(A∪C)
• A∩(B∪C)=(A∩B)∪(A∩C)

4. De Morgan's Laws:

• (A ∪ B)^c=A^c ∩B^c
• (A ∩ B)^c=A^c ∪ B^c

5. Cardinality Properties:

• n(A∪B)=n(A)+n(B)−n(A∩B)
• For three sets, n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(A∩C)+n(A∩B∩C)

7.0Solved Examples on Finite Set

Example 1: In a class of 50 students, 30 passed in Mathematics, 25 passed in Physics, and 10 passed in both. How many students failed in both subjects?

Solution: 

Let M be the set of students who passed in Maths and P be the set of students who passed in Physics.

n(M)=30, n(P)=25, n(M∩P)=10.

Total students n(U)=50.

First, find the number of students who passed in at least one subject:

 n(M∪P)=n(M)+n(P)−n(M∩P)

n(M∪P)=30+25−10=45

The number of students who failed in both subjects is the total number of students minus those who passed in at least one subject. 

Number of students who failed = n(U)−n(M∪P)=50−45=5.

Answer: 5 students failed in both subjects.

Example 2: In a survey of 100 people, it was found that 60 people read newspaper A, 40 read newspaper B, 30 read newspaper C, 20 read A and B, 15 read B and C, 10 read A and C, and 5 read all three newspapers. Find the number of people who read exactly two newspapers.

Solution: 

Let A,B,C be the sets of people who read newspapers A, B, and C respectively.

n(A)=60,n(B)=40,n(C)=30

n(A∩B)=20,n(B∩C)=15,n(A∩C)=10 

n(A∩B∩C)=5

Number of people who read exactly two newspapers can be found by summing up the numbers in the intersections and then subtracting the number of people who read all three (as they are counted in each intersection).

Number of people who read exactly two newspapers = (n(A∩B)−n(A∩B∩C))+(n(B∩C)−n(A∩B∩C))+(n(A∩C)−n(A∩B∩C))

=(20−5)+(15−5)+(10−5) 

=15+10+5=30

Answer: 30 people read exactly two newspapers.

8.0Practice Questions on Finite Set

1. Which of the following are finite sets?
   (a) The set of all even numbers less than 100
   (b) The set of natural numbers
   (c) The set of all vowels in the English alphabet
   (d) The set of integers greater than 0
2. If A = {2, 4, 6, 8, 10}, find the number of subsets of A.
3. Let B = {1, 2, 3, 4, 5, 6, 7}. Find the power set of B. How many elements does the power set contain?
4. State whether the following sets are finite or infinite:

• The set of all prime numbers less than 50
• The set of multiples of 7
• The set of all letters in the word "MATHEMATICS"

5. If a finite set has n elements, how many proper subsets does it have?
6. Consider C = $\{x\in \mathbb{N}:x^2\le 25\}$. Write the elements of set C. Is it finite or infinite?
7. A class has 40 students. Let S be the set of all students in the class. How many elements are there in the power set of S?