Sets Chapter-1 Class 11th Quick Revision Notes

Set
A set is a well-defined collection of objects.

Representation of Sets
There are two methods of representing a set

• Roster or Tabular form In the roster form, we list all the members of the set within braces { } and separate by commas.
• Set-builder form In the set-builder form, we list the property or properties satisfied by all the elements of the sets

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 belongs to A, we write a ∉ A.

Standard 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

(i) Roster/Listing Method/Tabular Form In this method, a set is described by listing element, separated by commas, within braces.

e.g., A = {a, e, i, o, u}

(ii) 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

1. Finite Set A set containing finite number of elements or no element.

2. 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).

3. Infinite Set A set containing infinite number of elements.

4. Empty/Null/Void Set A set containing no element, it is denoted by (φ) or { }.

5. Singleton Set A set containing a single element.

6. 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.

7. 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 true.

8. 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

9. 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.

10. Universal Set (U) A set consisting of all possible elements which occurs under consideration is called a universal set.

11. Comparable Sets Two sets A and Bare comparable, if A ⊆ B or B ⊆ A.

12. Non-Comparable Sets For two sets A and B, if neither A ⊆ B nor B ⊆ A, then A and Bare called non-comparable sets.

13. 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).

14. Disjoint Sets Two sets A and B are called disjoint, if, A ∩ B = (φ).

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.

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.

If A₁, A₂,… , An is a finite family of sets, then their intersection is denoted by

Complement of a Set

If A is a set with U as universal set, then complement of a set, denoted by A’ or Ac is the set U – A .

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.

Symmetric Difference

For two sets A and B, symmetric difference is the set (A – B) ∪ (B – A) denoted by A Δ B.

Laws of Algebra of Sets

For three sets A, B and C

(i) Commutative Laws
A ∩ B = B ∩ A
A ∪ B = B ∪ A

(ii) Associative Laws
(A ∩ B) ∩ C = A ∩ (B ∩ C)
(A ∪ B) ∪ C = A ∪ (B ∪ C)

(iii) Distributive Laws
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(iv) Idempotent Laws
A ∩ A = A
A ∪ A = A

(v) Identity Laws
A ∪ Φ = A
A ∩ U = A

(vi) De Morgan’s Laws

(a) (A ∩ B) ′ = A ′ ∪ B ′
(b) (A ∪ B) ′ = A ′ ∩ B ′
(c) A – (B ∩ C) = (A – B) ∩ (A- C)
(d) A – (B ∪ C) = (A – B) ∪ ( A – C)

(vii) (a) A – B = A ∩ B’
(b) B – A = B ∩ A’
(c) A – B = A ⇔A ∩ B= (Φ)
(d) (A – B) ∪ B= A ∪ B
(e) (A – B) ∩ B = (Φ)
(f) A ∩ B ⊆ A and A ∩ B ⊆ B
(g) A ∪ (A ∩ B)= A
(h) A ∩ (A ∪ B)= A

(viii) (a) (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B)
(b) A ∩ (B – C) = (A ∩ B) – (A ∩ C)
(c) A ∩ (B Δ C) = (A ∩ B) A (A ∩ C)
(d) (A ∩ B) ∪ (A – B) = A
(e) A ∪ (B – A) = (A ∪ B)

(ix) (a) U’ = (Φ)
(b) Φ’ = U
(c) (A’ )’ = A
(d) A ∩ A’ = (Φ)
(e) A ∪ A’ = U
(f) A ⊆ B ⇔ B’ ⊆ A’

Important Points to be Remembered

• 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. Suppose A = {1, 2}, thenP(A) = {{1}, {2},{1,2}, Φ}.(a) A ∉ P(A) (b) {A} ∈ P(A)
• (vii) If a set A has n elements, then P(A) or subset of A has 2_n elements.
• (viii) 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 Δ B) = n(A) + n(B)- 2n(A ∩ B)
• n(A ∪ B ∪ C)= n(A)+ n(B)+ n(C)- n(A ∩ B) – n(B ∩ C)- n(A ∩ C)+ n(A ∩ B ∩ C)
• n (number of elements in exactly two of the sets A, B, C) = n(A ∩ B) + n(B ∩ C) + n (C ∩ A)- 3n(A ∩ B ∩ C)
• n (number of elements in exactly one of the sets A, B, C) = n(A) + n(B) + n(C) – 2n(A ∩ B) – 2n(B ∩ C) – 2n(A ∩ C) + 3n(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)

Ordered Pair

An ordered pair consists of two objects or elements in a given fixed order.

Equality of Ordered Pairs Two ordered pairs (a1, b1) and (a2, b2) are equal iff a1 = a2 and b1 = b2.

Cartesian Product of Sets

For two sets A and B (non-empty sets), the set of all ordered pairs (a, b) such that a ∈ A and b ∈ B is called Cartesian product of the sets A and’ B, denoted by A x B. A x B={(a,b):a ∈ A and b ∈ B}

If there are three sets A, B, C and a ∈ A, be B and c ∈ C, then we form, an ordered triplet (a, b, c). The set of all ordered triplets (a, b, c) is called the cartesian product of these sets A, B and C.

i.e., A x B x C = {(a,b,c):a ∈ A,b ∈ B,c ∈ C}

Properties of Cartesian Product

For three sets A, B and C

• n (A x B)= n(A) n(B)
• A x B = Φ, if either A or B is an empty set.
• A x (B ∪ C)= (A x B) ∪ (A x C)
• A x (B ∩ C) = (A x B) ∩ (A x C)
• A x (B — C)= (A x B) — (A x C)
• (A x B) ∩ (C x D)= (A ∩ C) x (B ∩ D)
• If A ⊆ B and C ⊆ D, then (A x C) ⊂ (B x D)
• If A ⊆ B, then A x A ⊆ (A x B) ∩ (B x A)
• A x B = B x A ⇔ A = B
• If either A or B is an infinite set, then A x B is an infinite set.
• A x (B’ ∪ C’ )’ = (A x B) ∩ (A x C)
• A x (B’ ∩ C’ )’ = (A x B) ∪ (A x C)
²elements in common.
• If ≠ B, then A x B ≠ B x A
• If A = B, then A x B= B x A
• If A ⊆ B, then A x C = B x C for any set C.