How to Solve Sets Questions: Step-by-Step Guide
In mathematics, sets is defined as an organized collection of objects. Sets can be represented in set-builder form or roster form. Sets are usually represented in curly braces denoted by {}, for example, B = {1,2,3,4} is a set. Also, check the set symbols here.
In set theory, you will learn about sets maths set questions, set theory problems and solutions. It was developed to describe the collection of objects. The set theory generally defines the different types of sets, symbols, as well as operations performed.
What are the Elements of a Set?
Let us take an example:
B = {1, 2, 3, 4, 5 }
Since a set is usually represented by any capital letter. Thus, B is the set, and 1, 2, 3, 4, 5 are the elements of the set or members of the set. The elements that are written in the set can be in any order but these elements cannot be repeated. All the set elements are represented in small letters in the case of alphabets. Also, we can write it as 1 ∈ B, 2 ∈ B, etc. The cardinal number of the set B is 5. Some commonly used sets are as follows:
N: Set of all natural numbers
Z: Set of all integers
Q: Set of all rational numbers
R: Set of all real numbers
Z+: Set of all positive integers
What Does Order of Sets Mean?
The order of a set defines the number of elements in a set . It describes the size of a set. The order of the set is also known as the cardinality.
The size of a set whether it is is a finite set or an infinite set said to be a set of finite order or infinite order, respectively.
Representation of Sets
The sets are represented in curly braces, {}. For example, {2,3,4} or {a,b,c} or {Bat, Ball, Wickets}. The elements in the sets are depicted in either the Statement form,Set builder form or Roster Form.
Statement Form
The well-defined descriptions of a member of a set in statement form are written and enclosed in the curly brackets.
For example, a set of numbers that are even less than 15.
In statement form, it can be written as {even numbers less than the number 15}.
Roster Form
In Roster form, all the elements of any given set are listed.
For example, the set of natural numbers less than the number 5.
We know that natural number = 1, 2, 3, 4, 5, 6, 7, 8,……….
We know that Natural Number less than 5 = 1, 2, 3, 4
Therefore, the set is N = { 1, 2, 3, 4 }
Set Builder Form
The general form is, A equals { x : property }
Example: Write the following sets in set builder form: A={2, 4, 6, 8}
Solution:
2 = 2 x 1
4 = 2 x 2
6 = 2 x 3
8 = 2 x 4
So, the set builder form is A = {x: x=2n, n ∈ N and 1 ≤ n ≤ 4}
Also, Venn Diagrams are the best way for visualized representation of sets.
Maths Sets Questions
Maths Sets questions ( maths sets questions and answers) are given in this article for students to make them understand the concept easily. Practicing these set theory problems will help to go through the concept of sets theory problems. It is an important chapter for Class 11 students, hence we have given the questions based on the NCERT curriculum, with respect to the CBSE syllabus. Let’s discuss maths sets questions.
A brief introduction for each of the sub-topic of sets is also provided.
Here are some important definitions :
Sets: A collection of well-defined objects. It is denoted by Capital Letters.
Example: A = {1,2,3,4,5..}. Set A = collection of all natural numbers.
Roster Form : All elements are written in curly braces { }, separated by commas.
Example: R = {1, 3, 7, 21, 2, 6, 14, 42}.
Set Builder Form: The elements of the set represent a common property known as set builder form.
Example, R = {x : x is a vowel in English alphabet}
First, let us see some questions based on the representation of sets.
Maths Sets Questions
1. Let A and B be Two Given Finite Sets Such That n(A) Equals 20, n(B) Equals 28, and n(A ∪ B) Equals 36, Find n(A ∩ B).
Solution. Using the formula n(A ∪ B) equals n(A) + n(B) - n(A ∩ B), then n(A ∩ B) equals n(A) + n(B) - n(A ∪ B)
= 20 + 28 - 36
= 48 - 36
= 12
FAQs on Sets Questions: Definitions, Solutions & Practice
1. What are the two primary methods for representing a set as per the NCERT syllabus for Class 11? Provide an example for each.
According to the CBSE curriculum for the 2025-26 session, there are two main ways to represent a set:
- Roster or Tabular Form: In this method, all the elements of the set are listed, separated by commas, and enclosed within braces { }. For example, the set of the first five even natural numbers is represented as A = {2, 4, 6, 8, 10}.
- Set-Builder Form: In this method, we write a variable (like x) representing any member of the set followed by a property satisfied by each member of the set. For example, the same set A can be written as A = {x : x is an even natural number and x ≤ 10}.
2. What are some of the most frequently asked types of sets in exams? Briefly explain any four.
For Class 11 final exams, questions often test the understanding of various types of sets. Four important types are:
- Empty Set (or Null Set): A set containing no elements. It is denoted by ∅ or { }. For example, the set of whole numbers less than 0.
- Finite and Infinite Sets: A set is finite if it has a definite number of elements. It is infinite if the number of elements is not finite. For example, {1, 2, 3} is finite, while the set of all natural numbers {1, 2, 3, ...} is infinite.
- Equal Sets: Two sets A and B are equal if they have exactly the same elements, regardless of order. For example, if A = {1, 2, 3} and B = {3, 1, 2}, then A = B.
- Subsets: A set A is a subset of a set B if every element of A is also an element of B. It is denoted by A ⊂ B. For example, if A = {a, b} and B = {a, b, c}, then A is a subset of B.
3. Explain the key operations on sets—Union and Intersection—with an example. How are they represented in a Venn diagram?
Union and Intersection are fundamental operations on sets and are very important for exams.
The Union of two sets A and B, denoted by A ∪ B, is the set of all elements which are in A, or in B, or in both. In a Venn diagram, it represents the entire area covered by both circles.
The Intersection of two sets A and B, denoted by A ∩ B, is the set of all elements which are common to both A and B. In a Venn diagram, it represents the overlapping area of the two circles.
For example, if A = {1, 2, 3} and B = {3, 4, 5}, then:
A ∪ B = {1, 2, 3, 4, 5}
A ∩ B = {3}
4. From an exam perspective, what is the crucial difference between a subset and a proper subset?
The distinction between a subset and a proper subset is a common point of confusion and a potential trap in exam questions.
A set A is a subset of B (denoted A ⊆ B) if all elements of A are in B. This allows for the possibility that A and B are the same set (A = B).
However, A is a proper subset of B (denoted A ⊂ B) only if all elements of A are in B, and there is at least one element in B that is not in A. This means A must be strictly smaller than B (A ≠ B).
In short, every set is a subset of itself, but no set is a proper subset of itself. This is important when calculating the number of possible subsets (2ⁿ) versus the number of proper subsets (2ⁿ - 1).
5. How would you solve a typical 3-mark word problem involving the union of two sets?
A typical word problem involves a formula based on the Cardinality of Sets. To solve it, follow these steps:
1. Identify the given data: Read the problem carefully to find the number of elements in each set and their intersection. For instance, in a group of students, find n(A), n(B), and n(A ∩ B).
2. Identify what needs to be found: Determine if the question asks for the total number of elements in at least one set, which is the union n(A ∪ B).
3. Apply the formula: Use the principle of inclusion-exclusion: n(A ∪ B) = n(A) + n(B) - n(A ∩ B).
4. Substitute and Calculate: Plug in the known values into the formula and solve for the unknown. For example, if 20 students play cricket, 15 play football, and 5 play both, the total number of students playing at least one game is 20 + 15 - 5 = 30.
6. Why is the empty set (∅) considered a subset of every set? Explain the logic.
This is a fundamental concept in set theory. The definition of a subset is: 'A is a subset of B if there are no elements in A that are not in B'. Let's apply this to the empty set, ∅. To prove that ∅ is NOT a subset of some set S, we would need to find an element in ∅ that is not in S. But the empty set has no elements at all. Therefore, it's impossible to find an element in ∅ that isn't in S. Because this condition can never be met, the statement '∅ is a subset of S' is considered vacuously true for any set S.
7. What is a Power Set? If a set A has 4 elements, how many elements will its power set, P(A), have?
The power set of a set A, denoted as P(A), is the collection of all possible subsets of A, including the empty set and the set itself. This is a very important topic for 1-mark or 2-mark questions.
If a set A has 'n' elements, then its power set P(A) will have 2ⁿ elements.
In this case, set A has 4 elements (n=4).
Therefore, the number of elements in its power set P(A) will be 2⁴ = 16. This includes the empty set and the set A itself.
8. How do the concepts learned in the 'Sets' chapter form the basis for 'Relations and Functions'?
The chapter on Sets is foundational because it provides the language and tools for more advanced topics. The connection to 'Relations and Functions' is direct and critical:
- Cartesian Product: A relation is defined as a subset of the Cartesian product of two sets (A × B). Without understanding sets and subsets, you cannot define a relation.
- Domain and Range: The domain of a relation or function is the set of all first elements of the ordered pairs, and the range is the set of all second elements. These are defined using set theory.
- Types of Functions: Concepts like one-to-one and onto functions are explained by mapping elements from one set (the domain) to another set (the codomain).
