**NCERT CBSE NOTES CLASS 11 MATHS CHAPTER 1 **

**SETS**

**Chapter Notes**

**Definition of Set**

Set is a collection of well defined objects which are distinct from each other. Sets are usually denoted by capital letters A,B,C,… and elements are usually denoted by small letters *a,b,c,…*

If ‘*a*’ is an element of a set A, then we write *a* ∈ A and say ‘*a*’ belongs to A or ‘*a*’ is in A or ‘*a*’ is a member of A. If ‘*a*’ does not belong to A, we write *a* ∉ A.

**Standard Set Notations**

- N : A set of natural numbers.
- W : A set of whole numbers.
- Z : A set of integers.
- Z
^{+}/Z^{-}: A set of all positive/negative integers. - Q : A set of all rational numbers.
- Q
^{+}/Q^{-}: A set of all positive/ negative rational numbers. - R : A set of real numbers.
- R
^{+}/R^{-}: A set of all positive/negative real numbers. - C : A set of all complex numbers.

**Methods for Describing a Set**

**Roster/Listing Method/Tabular Form**

In this method, a set is described by listing elements,** **separated by commas, within braces. e.g., A = {*a*, e, i, o, u}

**Set Builder/Rule Method**

In this method, we write down a property or rule which gives us** **all the elements of the set by that rule. e.g., A = {*x* : *x* is a vowel of English alphabets}

**Types of Sets**

**Finite Set **

A set, containing finite number of elements or no element, is called finite set.** **

**Cardinal Number of a Finite Set **

The number of elements in a given finite set is called ** cardinal number** of finite set, denoted by

**n(A)**.

**Infinite Set **

A set containing infinite number of elements is called infinite set.** **

**Empty/Null/Void Set **

A set containing no element, it is denoted by Φ or { }.** **

**Singleton Set **

A set containing a single element, is called singleton set.** **

**Equal Sets **

Two sets A and B are said to be equal, if every element of A is a member of** **B and every element of B is a member of A and we write A = B.

**Equivalent Sets **

Two sets are said to be equivalent, if they have same number of** **elements.

If ** n(A) = n(B),** then A and B are equivalent sets. But converse is not necessarily true.

**Subset and Superset **

Let A and B be two sets. If every element of A is an element of B,** **then A is called subset of B and B is called superset of A. Written as

A ⊆ B or B ⊇ A

**Proper Subset**

If A is a subset of B and A ≠ B, then A is called proper subset of B and** **we write A ⊂ B.

**Universal Set (U)**

A set consisting all possible elements which occurs under** **consideration is called a universal set.

**Comparable Sets**

Two sets A and B are comparable, if A** **⊆** **B or B** **⊆** **A.** **

**Non-Comparable Sets **

For two sets A and B, if neither A** **⊆** **B nor B** **⊆** **A, then A and B** **are called non-comparable sets.

**Power Set (P) **

The set formed by all the subsets of a given set A, is called power set of** **A, denoted by P(A). If n(A) = m then **n(P(A)) = 2**^{m}.

**Disjoint Sets **

Two sets A and B are called disjoint, if, A ∩ B = (Φ).

**Intervals as subsets of R **

**Open Interval**

Let *a, b *∈ **R **and *a < b. *Then the set of real numbers { *y : a < y < b*} is called an *open interval *and is denoted by (*a*, *b*)*. *All the points between *a *and *b *belong to the open interval (*a, b*) but *a, b *themselves do not belong to this interval.

**Closed Interval**

The interval which contains the end points also is called *closed interval *and is denoted by [*a, b *].

[*a, b *] = {*x *: *a *≤ *x *≤ *b*}

- Intervals can be closed at one end and open at the other, i.e.,
[

*a, b)*= {*x : a*≤*x*<*b*} is an*open interval*from*a*to*b,*including*a*but excluding*b.*(

*a, b*] = {*x*:*a*<*x*≤*b*} is an*open interval*from*a*to*b*including*b*but excluding*a.*- The set [0, ∞) defines the set of non-negative real numbers, while set (– ∞, 0) defines the set of negative real numbers. The set (– ∞, ∞) describes the set of real numbers in relation to a line extending from – ∞ to ∞.

On real number line, various types of intervals described above as subsets of

**R**,**Venn Diagram**In a Venn diagram, the universal set is represented by a rectangular region and a set is represented by circle or a closed geometrical figure inside the universal set

**Operations on Sets****Union of Sets**The union of two sets A and B, denoted by A ∪ B is the set of all those elements, each one of which is either in A or in B or both in A and B.

A

**∪**B = {*x*:*x*∈ A or*x*∈ B }**Some Properties of the Operation of Union**- A ∪ B = B ∪ A (Commutative law)
- (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative law)
- A ∪ Φ = A (Law of identity element, Φ is the identity of ∪)
- A ∪ A = A (Idempotent law)
- U ∪ A = U (Law of U)

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

A ∩ B = {

*x*:*x*∈ A &*x*∈ B}If A and B are two sets such that A∩B = Φ, then A and B are called

*disjoint sets**.***Some Properties of Operation of Intersection**- A ∩ B = B ∩ A (Commutative law).

- (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative law).
- Φ ∩ A = Φ,
U ∩ A = A (Law of Φ and U).

- A ∩ A = A (Idempotent law).
- A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), i. e., ∩ distributes over ∪, and
A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C), i. e., ∪ distributes over ∩

(Distributive law)

**Complement of a Set**If A is a set with U as universal set, then the complement of A is the set of all elements of U which are not the elements of A. It is denoted by A’.

A′ = {

*x*:*x*∈ U and*x*∉ A }. ⇒ A′ = U – A**Some Properties of Complement Sets****1.**Complement laws:(i) A ∪ A′ = U (ii) A ∩ A′ = Φ

**2.**De Morgan’s law:(i) (A ∪ B)´ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′

**3.**Law of double complementation: (A′)′ = A**4.**Laws of empty set and universal setΦ′ = U and U′ = Φ.

**Difference of Sets**For two sets A and B, the difference A-B is the set of all those elements of A which do not belong to B.

The sets A-B, A∩B and B-A are mutually disjoint sets, i.e., the intersection of any of these two sets is the null set

**Symmetric Difference of Sets**For two sets A and B, symmetric difference is the set (A-B) ∪ (B-A) denoted by A~B.

**Important Points to be Remembered About Sets**- Every set is a subset of itself i.e., A ⊆ A, for any set A.
- Empty set Φ is a subset of every set i.e., Φ ⊂ A, for any set A.
- For any set A and its universal set U, A ⊆ U
- If A = Φ, then power set has only one element i.e., n(P(A)) = 1
- Power set of any set is always a non-empty set.
- If a set A has m elements, then P(A) or power set of A has 2
^{m}elements. - Equal sets are always equivalent but equivalent sets may not be equal.
- The set {Φ} is not a null set. It is a set containing one element Φ.

**Results on Number of Elements in Sets**- n (A∪B) = n(A) + (B) - n(A∩B)
- n(A∪B) = n(A) + n(B), if A and B are disjoint.
- n(A – B) = n(A) – n(A ∩ B)
- n(A + B) = n(A) + n(B) - 2n(A ∩ B) [Belonging to exactly one]
- n(A ∪ B ∪ C)= n(A) + n(B) + n(C) - n(A ∩ B) – n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C)
- n(A’ ∪ B’) = n(A ∩ B)’ = n(U) – n(A ∩ B)
- n(A’ ∩ B’) = n(A ∪ B)’ = n(U) – n(A ∪ B)
- n(B – A) = n(B) - n(A ∩ B)