Practice Problems on Finite and Infinite Sets

Finite and infinite sets are two different parts of “Set theory” in mathematics. When a set has a finite number of elements, it is called a “Finite set” and when a set has an infinite number of elements, it is called an “Infinite set”. A finite set is countable, whereas an infinite set is uncountable. The elements in a finite set are natural numbers i.e., non-negative integers. We use dots in a set to represent an infinite set. In this articles we will discuss about the finite and infinite sets, their properties, solved problems and practice questions.

Finite Sets

A finite set has a finite or countable number of elements. It is a set of natural numbers i.e., positive integers and can be easily counted. It is expressed as P = {1, 2, 3, 4, . . ., n} for natural number n. For example, {5, 6, 7, 8} is a set of countable numbers. An empty set { } is also considered a finite set as it has zero elements i.e., P={ } or n(A) = 0.

Example of Fine Sets

• A set of days in a week: ( Sunday, Monday, Tuesday, Wednesday, Thursday, Friday and Saturday )
• The set of natural numbers less than 10: {1, 2, 3, 4, 5, 6, 7, 8, 9}
• The set of vowels in the English alphabet: {a, e, i, o, u}
• The set of planets in the Solar System: {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}

Infinite Set

An infinite set, as the name suggests, has an infinite or uncountable number of elements. It is a set of all whole numbers including non-negative and negative integers. Infinite sets are also known as uncountable sets. We know that two infinite sets always form an infinite set. It is expressed as P = {0, 1, 2, 3, . . . }, a set of all the whole numbers.

Some examples of Infinite Sets are:

• Set of whole numbers
• Set of integers
• Line segments in a plane etc.

Problem 1: Let A= {1,2,3,4} and C= {4,5,6,8} are finite sets. Find the union and intersection of sets A and C.

Solution:

• The union of two sets A and C can be written as A∪C. It is the set of all the unique elements of both the sets A and C.

Therefore, A∪C = {1,2,3,4,5,6,8}.

• The intersection of the two sets A and C is written as A∩C, is the set of elements which are common in both the sets.

Therefore, A∩C = {4}.

Problem 2: Let P = {1,2,3,4,5,6} be a finite set. How many subsets does set P have?

Solution:

• To calculate the subsets of a set, we will use the formula 2ⁿ , where n= the number of elements in the set.

Now, in this question, n= 5

The number of subsets = 2ⁿ =2⁵ = 32.

Therefore, the set P has 32 subsets including the empty set and the set itself.

Problem 3: State whether the given set is finite or infinite?

1) {1,3,5, . . . }

2) {2, 3, 4, 5}

3) { . . . , -3, -2, -1, 0, 1, 2, 3}

4) {10, 20, 30, . . . 200}

Solution:

1) Infinite

2) Finite

3) Infinite

4) Finite

Problem 4: State whether the given set is finite or infinite?

1) {0}

2) {Φ}

3) {X │X∈ N and X>10}

4) { X │X is a prime number}

Solution:

1) Finite

2) Finite

3) Infinite

4) Infinite

Problem 5: If Tan θ = 1, then the solution set of the equation is finite or infinite?

Solution:

Given that tan θ = 1

⇒ tan θ = 1 = tan ?/4

⇒ θ = n? + ?/4 , where n∈Z

⇒ θ = { X : X∈ (n? + ?/4), n∈Z }

The solution of the given question is an infinite set.

Problem 6: Consider the following statement:

1) P = {X : X is prime, such that 1> X >10} and A = {2,3,5,7} are equal sets.

2) P = {a,e,i,o,u} and Q = {a,i,o,e,u} are unequal sets.

Which of the following statement is correct?

Solution:

1) Given, X is prime such that 1> X >10

hence X = 2,3,5,7

P = {2,3,5,7} = A⇒ P=A

Therefore both the sets are equal and the statement is correct.

2) Given, P = {a,e,i,o,u} and Q = {a,i,o,e,u} are unequal sets.

where the order of the elements in a set does not impact the equality of the two sets.

Hence, set P and set Q are equal and the given statement is incorrect.

Problem 7: Find out whether the following set is finite or not.

A = {X : X∈N and (X-1) (X-2) = 0}

Solution:

Given (X-1) (X-2)=0

X= 1,2 then the given set is= (1,2)

As the set has two elements and is countable,

Hence, the set is finite.

Problem 8: Identify the set is finite or infinite?

(1, 1/2, 1/4, 1/8, . . . )

Solution:

The given set is infinite as it has no end point within the set.

Problem 9: Find out whether the following site is finite or infinite?

A = {X: X ∈ N and X² = 4}

Solution:

Given, X² = 4

⇒ X= +2 or -2

⇒ X = 2,-2 but X is a natural number, it can not be negative, hence X = 2

therefore A = (2) which is countable. The set is a finite set.

Read More,

• Set theory
• Finite sets
• Infinite sets

A few practice questions for the students are given below:

Q1: A= {X:X∈R such that X²-7X+12 = 0}, then A is a finite or an infinite set?

Q2: Let A = {1,2,3,4,5} and B= {4,5,6,7,8}, then A∩B is finite or infinite?

Q3: Let A = {2, 4, 6, 8, 10} and B = {1, 3, 5, 7, 9} be two finite sets. Find the cardinality of the A∪B.

Q4: let you have two finite sets that have m and n elements, in how many ways you can create a new set by combining elements from both the sets?

Q5: Let A ={2,4,6,8,10}, how many subsets does the set A have?

Q6: Check whether the given sets are finite or infinite:

• Factors of 25 ,
• Multiples of 2,
• Lines segments in a plane

Q7: Proof the power set of set B = {3, 5, 7 . . . } is an infinite set.

Q8: Check whether the given sets are finite or infinite:

• P = {2, 4, 6, . . . }
• M = {1, 2, 3, 4, 5}
• X = {a, b, c, . . . }

What is finite set in set theory?

When a set has finite number of elements, it is called a “Finite set”. It is a set of natural numbers i.e., positive integers and can be easily counted.

How do you determine cardinality of a finite set?

The number of elements in a finite set is known as the cardinality of that set. The cardinality of a finite set is expressed as n(A).

Are all subsets of an finite set a finite set?

Yes, every subset of a finite set is also finite. The subsets are determined by using the formula 2ⁿ .

How to determine if a set is finite or infinite?

A finite set has a starting and ending point, whereas an infinite set is endless from both the starting and ending. If a set consists countable number of elements then it is a Finite set and if the elements are uncountable then it is an Infinite set.

Is 0 (zero) infinite or finite set?

Zero is a finite number. We know that the cardinality of an empty set is zero and also zero represents the absence of null quantity of something, so, we can say that zero is a finite set.

Is there any symbol for finite set?

No, there is no particular symbol for representing a finite set. It is represented as a normal set with alphabets like A, M, S etc. For example: A = { 1,2,3,4} is a finite set.

Can you represent an infinite set Roster form?

Infinite sets can not be represented in roster form.