Sets Class 11 Notes: CBSE Maths Chapter 1

1. Sets

A set is a collection of well-defined and well distinguished objects of our perception or thought.

Notations

The sets are usually denoted by capital letters A, B, C, etc. and the members or elements of the set are denoted by lower- case letters a, b, c, etc. If x is a member of the set A, we write $x \in A$(read as 'x belongs to A') and if x is not a member of the set A, we write $x \notin A$ (read as 'x does not belong to A’). If x and y both belong to A, we write $x,y \in A$.

Usually, sets are represented in the following two ways:

(i) Roster form or Tabular form

(ii) Set Builder form or Rule Method

(i) Roster form: In this form, we list all the members of the set within braces (curly brackets) and separate these by commas. For example, the set A of all odd natural numbers less than 10 in the Roster form is written as:

A = {1, 3, 5, 7, 9}

(a) In roster form, every element of the set is listed only once.

(b) The order in which the elements are listed is immaterial. For example, each of the following sets denotes the same set {1, 2, 3}, {3, 2, 1}, {1, 3, 2}.

(ii) Set- Builder Form

In this form, we write a variable (say x) representing any member of the set followed by a property satisfied by each member of the set.

For example, the set A of all prime numbers less than 10 in the set-builder form is written as

A = {x | x is a prime number less than 10}

The symbol '|' stands for the words 'such that'. Sometimes, we use the symbol ':' in place of the symbol '|'.

2. Types of Sets

(i) Empty Set or Null Set: A set which has no element is called the null set or empty set. It is denoted by the symbol $\phi $.

For example, each of the following is a null set:

(a) The set of all real numbers whose square is –1.

(b) The set of all rational numbers whose square is 2.

(c) The set of all those integers that are both even and odd.

A set consisting of at least one element is called a non-empty set.

(ii) Singleton Set: A set having only one element is called a singleton set.

For example, {0} is a singleton set, whose only member is 0.

(iii) Finite and Infinite Sets:

A set which has a finite number of elements is called a finite set. Otherwise, it is called an infinite set.

For example, the set of all days in a week is a finite set whereas the set of all integers, denoted by

{.... -2, -1, 0, 1, 2...} or {x | x is an integer}, is an infinite set. An empty set $\phi $ which has no element in a finite set A is called empty or void or null set.

(iv) Cardinal Number: The number of elements in finite set is represented by n(A), known as Cardinal number.

(v) Equal Sets: Two sets A and B are said to be equals, written as A = B, if every element of A is in B and every element of B is in A.

(vi) Equivalent Sets: Two finite sets A and B are said to be equivalent, if n (A) = n (B). Clearly, equal sets are equivalent but equivalent sets need not be equal.

For example, the sets A = {4, 5, 3, 2} and B = {1, 6, 8, 9} are equivalent but are not equal.

(vii) Subset: Let A and B be two sets. If every element of A is an element of B, then A is called a subset of B, and we write $A \subset B$ or $B \supset A$ (read as 'A is contained in B' or B contains A'). B is called superset of A.

• Every set is a subset and a superset itself.

• If A is not a subset of B, we write \[A \not\subset B\].

• The empty set is the subset of every set.

• If A is a set with n(A) = m, then the number of subsets of A are 2m and the number of proper subsets of A are 2m -1.

For example, let A = {3, 4}, then the subsets of A are $\phi $, {3}, {4}, {3, 4}. Here, 

n(A) = 2 and number of subsets of $A = {2^2} = 4$. Also, {3}$ \subset ${3,4}and {2,3}$ \not\subset ${3, 4}.

(viii) Power Set: The set of all subsets of a given set A is called the power set of A and is denoted by P(A).

For example, if A = {1, 2, 3}, then

P(A) = { $\phi $, {1}, {2}, {3}, {1,2} {1, 3}, {2, 3}, {1, 2, 3}}

Clearly, if A has n elements, then its power set P (A) contains exactly $2n$ elements.

Operations on Sets

3. Union of Two Sets: The union of two sets A and B, written as A u B (read as 'A union B'), is the set consisting of all the elements which are either in A or in B or in both.

Thus, \[A \cup B{\text{ }} = {\text{ }}\left\{ {x{\text{ }}:{\text{ }}x{\text{ }} \in {\text{ }}A{\text{ }}or{\text{ }}x{\text{ }} \in B} \right\}\]

Clearly, \[x \in A \cup B \Rightarrow x \in A{\text{ }}or{\text{ }}x \in B\], and \[x \notin A \cup B \Rightarrow x \notin A{\text{ }}and{\text{ }}x \notin B.\]

For example, if A = {a, b, c, d} and B = {c, d, e, f}, then $A \cup B$ = {a, b, c, d, e, f}.

4. Intersection of two sets: The intersection of two sets A and B, written as A n B

(read as ‘A’ intersection ‘B’) is the set consisting of all the common elements of A and B. Thus,

\[A \cap B{\text{ }} = {\text{ }}\left\{ {x:x \in A{\text{ }}and{\text{ }}x \in B} \right\}\]

Clearly, \[x \in A \cap B \Rightarrow x \in A{\text{ }}and{\text{ }}x \in B,{\text{ }}and{\text{ }}x \notin A \cap B \Rightarrow x \notin A{\text{ }}or{\text{ }}x \notin B\]

For example, if A = {a, b, c, d} and B = {c, d, e, f}, then $A \cap B$ = {c, d}.

5. Disjoint Sets: Two sets A and B are said to be disjoint, if\[A \cap B{\text{ }} = \phi \], i.e. A and B have no element in common.

For example, if A = {1, 3, 5} and B = {2, 4, 6}, then \[A \cap B{\text{ }} = {\text{ }}\phi \], so A and B are disjoint sets.

6. Difference of two sets: If A and B are two sets, then their difference A-B is defined as:

\[A - B{\text{ }} = {\text{ }}\left\{ {x{\text{ }}:{\text{ }}x \in A{\text{ }}and{\text{ }}x \notin B} \right\}\]. Similarly, \[B - A{\text{ }} = {\text{ }}\left\{ {x{\text{ }}:{\text{ }}x \in B{\text{ }}and{\text{ }}x \notin A{\text{ }}} \right\}.\]

For example, if A= {1, 2, 3, 4, 5} and B = {1, 3, 5, 7, 9} then A - B = {2, 4} and \[B - A{\text{ }} = {\text{ }}\left\{ {7,9} \right\}.\]

Important Results:

(a) A – B $ \ne $ B – A

(b) The sets A – B , B – A and $A \cap B$ are disjoint sets

(c) A – B $ \subset $ A and B – A $ \subset $ B

(d) $A - \phi  = A$and \[A - A{\text{ }} = \phi \]

7. Symmetric Difference of Two Sets: The symmetric difference of two sets A and B, denoted by A ∆ B, is defined as

A ∆ B = (A – B) $ \cup $(B – A).

For example, if A= {1,2,3,4,5} and B = {1,3,5,7,9} then,

A ∆ B = (A– B) $ \cup $ (B – A) = \[\left\{ {2,4} \right\} \cup \left\{ {7,9} \right\}\]= {2,4,7,9}.

8. Complement of a Set: If U is a universal set and A is a subset of U, then the complement of A is the set which contains those elements of U, which are not contained in A and is denoted by $A'$or ${A^c}$. Thus,

${A^c} = \left\{ {x{\text{ }}:{\text{ }}x \in U{\text{ }}and{\text{ }}x \notin A} \right\}$ 

For example, if U = {1,2,3,4 ...} and A = {2,4,6,8...} then, ${A^c}$= {1,3,5,7, ...}

Important Results

a) \[{U^c} = \phi \]

b) \[{\phi ^c} = {\text{ }}U\]

c) \[A \cup {A^c} = {\text{ }}U\]

d) \[A \cap {A^c} = {\text{ }}\phi \]

9. Algebra of Sets

1. For any set A, we have 

a) \[A \cup A = A\]

b) \[A \cap A = A\]

2. For any set A, we have 

c) \[A \cup \phi  = A\]

d) \[A \cap \phi  = \phi \]

e) \[A \cup U = U\]

f) \[A \cap U = A\]

3. For any two sets A and B, we have 

g) \[A \cup B = B \cup A\]

h) \[A \cap B = B \cap A\]

4. For any three sets A, B and C, we have 

i) \[A \cup (B \cup C) = (A \cup B) \cup C\]

j) \[A \cap \left( {B \cap C} \right) = \left( {A \cap B} \right) \cap C\]

5. For any three sets A, B and C, we have 

k) \[A \cup \left( {B \cap C} \right) = (A \cup B) \cap (A \cup C)\]

l) \[A \cap (B \cup C) = \left( {A \cap B} \right) \cup \left( {A \cap C} \right)\]

6. If A is any set, we have\[{\left( {{A^c}} \right)^c} = A\].

7. De Morgan's Laws For any three sets A, B and C, we have 

m) \[{(A \cup B)^c} = {A^c} \cap {B^c}\] 

n) \[{\left( {A \cap B} \right)^c} = {A^c} \cup {B^c}\]

o) \[A - {\text{ }}(B \cup C) = \left( {A - B} \right) \cap \left( {A - C} \right)\]

p) \[A - {\text{ }}(B \cap C) = \left( {A - B} \right) \cap \left( {A - C} \right)\]

Important Results on Operations on Sets

(i) \[A \subseteq A \cup B,{\text{ }}B \subseteq A \cup B,{\text{ }}A \cap B \subseteq A,{\text{ }}A \cap B \subseteq B\]

(ii) \[A - B = A \cap \;{B^c}\]

(iii) \[\left( {A - B} \right) \cup B = A \cup B\]

(iv) \[\left( {A - B} \right) \cap B = \phi \]

(v) \[A \subseteq B \Leftrightarrow {B^c} \subseteq {A^c}\]

(vi) \[A - B = {B^c} - {A^c}\]

(vii) \[(A \cup B) \cap (A \cup {B^c}) = A\]

(viii) \[A \cup B{\text{ }} = {\text{ }}\left( {A - B} \right) \cup \left( {B - A} \right) \cup \left( {A \cap B} \right)\]

(ix) \[A - \left( {A - B} \right){\text{ }} = {\text{ }}A \cap B\]

(x) \[A - B = B - A \Leftrightarrow A = B\]

(xi) \[A \cup B = A \cap B \Leftrightarrow A = B\]

(xii) \[A \cap \left( {B\Delta C} \right){\text{ }} = {\text{ }}\left( {A \cap B} \right)\Delta \left( {A \cap C} \right)\]

Important Formulas of Class 11 Maths Chapter 1 Sets You Shouldn’t Miss!

For Any Three Sets A, B, C.

1. n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

2. If A ∩ B = { }, then n(A ∪ B) = n(A) + n(B)

3. n(A - B) + n(A ∩ B) = n(A)

4. n(B - A) + n(A ∩ B) = n(B)

5. n(A ∪ B) = n(A - B) + n(A ∩ B) + n(B - A)

6. n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n(A ∩ B ∩ C)

Importance of Revision Notes

• Foundational Knowledge: Class 11 Maths Chapter 1 Sets Notes PDF introduces essential concepts like types of sets, Venn diagrams, and set operations, which are fundamental for further study in mathematics.

• Convenient Learning: The Sets Class 11 Maths Notes PDF download offers easy access to comprehensive notes, allowing for flexible and on-the-go study.

• Effective Revision: With clear explanations and examples, the notes support efficient revision and a deeper understanding of key concepts.

• Enhanced Exam Preparation: The notes help consolidate knowledge, improving readiness for exams and boosting overall academic performance.

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Tips for Learning the Class 11 Maths Chapter 1 Sets

Here are some tips for learning Class 11 Maths Chapter 1 Sets:

• Understand Basic Concepts: Start by thoroughly understanding the fundamental concepts of sets, including types of sets (finite, infinite, equal, etc.), set notation, and Venn diagrams.

• Practise Venn Diagrams: Use Venn diagrams to visualise and solve problems related to unions, intersections, and differences of sets. This will help you grasp how sets interact with each other.

• Work Through Examples: Go through various examples provided in the NCERT textbook and notes. Solve different types of problems to become familiar with various approaches and solutions.

• Use Practice Exercises: Regularly complete practice exercises and sample questions. This helps reinforce your understanding and improves problem-solving skills.

• Review Definitions and Properties: Make sure you memorise important definitions and properties of sets, such as the laws of set operations and De Morgan’s laws.

Conclusion

The Class 11 Maths Chapter 1 Sets Notes are a valuable resource for students aiming to master the foundational concepts of set theory. With comprehensive explanations, solved examples, and practice questions, these notes are designed to support effective learning and exam preparation. Whether you are downloading the Class 11 Maths Chapter 1 Sets Notes PDF for FREE or seeking structured guidance, these notes are essential for a thorough understanding of the chapter. Regular review of the Class 11 Maths Chapter 1 Sets Notes will enhance your problem-solving skills and ensure you are well-prepared for your exams.

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