All Formulas of Sets Class 11
Sets are one of the integral parts of Class 11 mathematics. It introduces the set theory which is one of the basics for higher studies. Hence, the chapter is important and mustn't be neglected. The chapter is also slightly formula based and hence sets of formulas Class 11 are necessary. For this reason, here we are presenting to you all the important sets of Class 11 Formulas. We will be discussing all the important formulas and with that, we'll also see the significance of each of the formulas.
Our expert faculty prepares the NCERT Solutions For Class 11 Maths Chapter 1 Sets following the latest CBSE Syllabus for 2021-22 as applied to 2019. The NCERT Solutions in Maths provide students with an effective and efficient process of solving problems. more effectively and efficiently. Furthermore, we focus on creating step-by-step solutions for all NCERT problems in a format that is easy to understand for students.
In Chapter 1 of the NCERT textbook, sets are used to define concepts of functions and relations. Class 11 Maths Chapter 1 from NCERT is devoted to a concept known as I. It contains basic definitions and operations related to sets. Since sequences, geometry, and probability are all built on Sets, it is essential to have a solid foundational knowledge of them. However, it is among the most straightforward chapters in NCERT Class 11 Maths to score maximum marks. These Vedantu NCERT Solutions are helpful for students who are looking for a simple and quick way of resolving questions.
What is a Set?
A set is a defined collection of objects. A set that contains a definite number of objects is called a definite set. Whereas a set consisting of an indefinite number of elements is called indefinite sets.
Example of a finite set: {1,2,3,4}.
Example of an infinite set: the set of all the natural numbers.
Now to understand all the formulas, firstly let us understand all the symbols used and what they signify.
When you want to unify or add two sets A and B, it is represented through A U B. Finding the union of two sets gives us a set containing all the elements contained in both A and B.
When you want to find the common elements between two sets A and B then, you need to find the set A intersection B or A inverted U B.
Sets can be added and subtracted.
You can also find A bar, this shall give you all the elements which are not contained in the set. This is called the complement of the set.
What is the Cardinality of a Set?
The cardinality of a set can be defined as the number of elements contained in a set. It could range from 0 to infinity.
For instance
Consider the set A = {1,2,3,4}.
The cardinality of set A is represented as n(A), which is 4 since A contains 4 elements.
Let's take another example for a better understanding:
Now consider the set of all the integers Z.
What would the cardinality of Z be?
Well, we do not know the number of integers hence the cardinality of the set Z would be indefinite.
Consider two sets of unknown cardinality:
A and B.
n(AᴜB) represents the total number of elements present in both of the sets A and B combined.
n(AᴜB) = n(A) + (n(B) – n(A∩B).
The Above-mentioned Formula is one of the most important formulae of set theory. It is used to find AᴜB. Let's understand this Formula in detail using a Venn diagram.
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The diagram given above is called the Venn diagram. It represents two different sets A and B. The region of the Venn diagram which is highlighted in pink are the elements that are common to both sets A and B. If we simply add the elements of A and B together, the elements belonging to the pink part would be added twice and hence will give us an incorrect sum. Hence while finding the union of two sets we need to subtract the intersection of the two sets one time. This will negate the error caused earlier and find the perfect union between the two sets.
Now, let us jump to the next level and try understanding the formula for 3 different sets:
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Consider 3 sets intersecting like in the Venn diagram given above.
n(AUBUC) = n(A) + n(B) + n(C) – n(A⋂B) – n(B⋂C) – n(C⋂A) + n(A⋂B⋂C)
From the image, we can visualize that if we simply add the sets together, some of the regions will be added multiple times. Hence, using this Formula we rectify the error and subtract the regions that have been added multiple times.
What Differentiates NCERT Solutions for Class 11 Maths Chapter 1- Sets from Others
This chapter consists of six exercises and a miscellaneous exercise designed to help students understand the concepts related to Sets of Class 11 Maths CBSE Syllabus (2021-22) thoroughly. NCERT Solutions for Class 11 Maths explains the following topics in Chapter 1:
A set is a collection of objects with a well-defined structure.
When there is no element in a set, then it is called an empty set.
Sets with definite numbers of elements are defined as finite sets
Sets consisting of infinite elements are defined as infinite setsIf two sets A and B contain the same elements, then they are considered equal.
When every element of a given set A is also an element of another set B, a set A is said to be a subset of another set B. R can be broken down into intervals.
There are multiple subsets of a set A which comprise a power set. They are indicated by P(A).
Those elements that are either in A or B are found in the set A+B, which is the union of two sets A and B.
A set of all elements which are common to two different sets A and B is the intersection of the two. Those elements that belong to A, but not to B, make up the difference between two sets A and B in this order.
Among all the elements of a universal set U, the complement of a subset A is a set of all elements that are not also elements of A.
A and B are equivalent. That is, (A + B)' = A′ * B′ and (A + B)' = A'*B'.
Solved Examples Sets Class 11 Maths
Example 1: In a school, there are 200 children, 65 like drawing and 85 like music. 25 like both. Find out how many of them like either of them or neither of them?
Solution:
Total number of children, n(μ) = 200
Number of children who enjoy drawing, n(d) = 65
Number of children who enjoy music, n(m) = 85
Number of students who like both, n(d∩m) = 25
Number of students who like either of them,
n(dᴜm) = n(d) + n(m) – n(d∩m)
→ 65 + 85 - 25 = 125
Number of students who like neither = n(μ) – n(dᴜm) = 200 – 150 = 50.
FAQs on CBSE Class 11 Maths Sets Formulas
1. What is the fundamental formula for the union of two sets in CBSE Class 11 Maths?
The fundamental formula for finding the number of elements in the union of two finite sets, A and B, is the Principle of Inclusion-Exclusion. The formula is: n(A U B) = n(A) + n(B) – n(A ∩ B). Here, n(A U B) is the number of elements in either set A or set B, n(A) and n(B) are the number of elements in each set respectively, and n(A ∩ B) is the number of elements common to both sets. We subtract the intersection to avoid counting the common elements twice.
2. How are set formulas applied to solve real-world problems?
Set formulas are used to analyse survey data and solve practical counting problems. For example, if a survey of 200 people finds 120 like coffee (C) and 90 like tea (T), with 40 liking both, we can find how many like at least one drink. Using the formula: n(C U T) = n(C) + n(T) - n(C ∩ T).
So, n(C U T) = 120 + 90 - 40 = 170. This means 170 people like at least one of the two beverages. To find those who like neither, we would subtract this from the total: 200 - 170 = 30 people.
3. What is the difference between the formula for the union of disjoint sets and overlapping sets?
The key difference lies in the intersection of the sets.
- For overlapping sets (which share common elements), the intersection n(A ∩ B) is a non-zero value. Thus, the full formula must be used: n(A U B) = n(A) + n(B) – n(A ∩ B).
- For disjoint sets (which have no common elements), their intersection is an empty set, meaning n(A ∩ B) = 0. The formula simplifies to: n(A U B) = n(A) + n(B).
4. Why is it important to understand Venn diagrams alongside set formulas?
Simply memorising formulas is not enough; understanding them through Venn diagrams is crucial for true comprehension. Venn diagrams provide a visual representation of the relationships between sets, making the logic behind formulas like the Principle of Inclusion-Exclusion intuitive. They show exactly why the intersection is subtracted to prevent double-counting. This visual understanding helps in tackling complex or non-standard problems where direct formula application might not be obvious.
5. How is the union formula for two sets extended to three sets (A, B, and C)?
The Principle of Inclusion-Exclusion for three sets (A, B, and C) is an extension of the two-set formula. The formula is:
n(A U B U C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C).
This works by adding the sizes of all three sets, subtracting the sizes of all pairwise intersections to correct for overcounting, and finally adding back the size of the three-way intersection, which was subtracted one too many times.
6. What are De Morgan's Laws and what is their importance in set theory?
De Morgan's Laws are fundamental formulas that relate the union and intersection of sets with their complements. The two laws are:
- (A U B)' = A' ∩ B': The complement of the union is the intersection of the complements.
- (A ∩ B)' = A' U B': The complement of the intersection is the union of the complements.
7. What are some key types of sets a student must know before applying formulas in Class 11 Maths?
Before applying formulas, it's essential to understand the basic types of sets as per the CBSE syllabus. These include:
- Empty Set (or Null Set): A set with no elements, denoted by {} or ∅.
- Singleton Set: A set with only one element.
- Finite and Infinite Sets: Sets with a countable or uncountable number of elements, respectively.
- Subset and Superset: If all elements of set A are in set B, A is a subset of B.
- Universal Set (U): The set containing all possible elements under consideration for a specific problem.
- Power Set: The set of all possible subsets of a given set.
