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Sets | Notes | Math XI
Sets
Key Concepts
- A set is a well-defined collection of objects.
- Sets can be represented in two ways — Roster or Tabular form and Set Builder form.
- Roster form: All the elements of a set are listed and separated by commas and are enclosed within braces { }. Elements are not repeated generally.
- Set Builder form: In set builder form, a set is denoted by stating the properties which its members satisfy.
- A set does not change if one or more elements of the set are repeated.
- An empty set is the set having no elements in it. It is denoted by φ or { }.
- A set having a single element is called a singleton set.
- On the basis of the number of elements, sets are of two types — finite and infinite sets.
- A finite set is a set in which there are a definite number of elements. Now, φ or { } or null set is a finite set as it has 0 number of elements, which is a definite number.
- A set which is not finite is called an infinite set.
- All infinite sets cannot be described in the roster form.
- Two sets are equal if they have exactly the same elements.
- Two sets are said to be equivalent if they have the same number of elements.
- Set A is a subset of set B if every element of A is in B, i.e. there is no element in A which is not in B and is denoted by A ⊆ B.
- A is a proper subset of B if and only if every element in A is also in B and there exists at least one element in B that is not in A.It is denoted by A ⊂ B.
- If A is a proper subset of B, then B is a superset of A and is denoted by B ⊇ A.
- Let A be a set. Therefore, the collection of all subsets of A is called the power set of A and is denoted by P(A).
- Common set notations
- N: The set of all natural numbers
- Z: The set of all integers
- Q: The set of all rational numbers
- R: The set of real numbers
- Z+: The set of positive integers
- Q+: The set of positive rational numbers
- R+: The set of positive real numbers
- N ⊂ R , Q ⊂ R , Q ⊂ Z, R ⊂ Z and N ⊂ R+
- If A ⊆ B and B ⊆ A, then A = B.
- Null set φ is the subset of every set including the null set itself.
- Open interval: The interval which contains all the elements between a and b excluding a and b.
In set notations: (a, b) = { x: a < x < b}

Closed interval: The interval which contains all the elements between a and b and also the end points a and b is called the closed interval.
[a, b] = {x: a ≤ x ≤ b}

- Semi-open intervals
[a, b) = {x: a ≤ x < b} includes all the elements from a to b including a and excluding b.

(a, b] = {x: a < x ≤ b} includes all the elements from a to b excluding a and including b.

- Universal set refers to a particular context.
It is the basic set which is relevant to that context. The universal set is usually denoted by U.
- Union of sets A and B, denoted by A ∪ B is defined as the set of all the elements which are either in A or B or both.
- Intersection of sets A and B, denoted by A ∩ B, is defined as the set of all the elements which are common to both A and B.
- The difference of the sets A and B is the set of elements, which belong to A but not to B and it is written as A – B and read as ‘A minus B’.
In set notations A – B = {x: x ∈ A, x ∉ B} and B – A = {x: x ∈ B, x ∉ A}.
- If the intersection of two non-empty sets is empty, i.e. A ∩ B = φ, then A and B are disjoint sets.
- Let U be the universal set and A be a subset of U. Thus, the complement of A, written as A’ or Ac, is the set of all elements of U which are not in set A.
- The number of elements present in a set is known as the cardinal number of the set or cardinality of the set. It is denoted by n(A).
- If A is a subset of U, then A’ is also a subset of U.
- Counting theorems are together known as the Inclusion–Exclusion Principle. This principle helps in determining the cardinality of union and intersection of sets.
- Sets can be represented graphically using Venn diagrams. Venn diagrams consist of rectangles and closed curves, usually circles. The universal set is generally represented by a rectangle and its subsets by circles.

Key Formulae
- Union of sets
A ∪ B = {x: x ∈ A or x ∈ B } ∪∩∈∉

- Intersection of sets
A ∩ B = {x: x ∈ A and x ∈ B }

- Complement of a set
A’ = {x: x ∈ U and x ∉ A}, A’ = U – A

- Difference of sets
A – B = {x: x ∈ A, x ∉ B} and B – A = {x: x ∈ B, x ∉ A}

- Symmetric difference of two sets:
Let A and B be two sets.
Hence, the symmetric difference of sets A and B is the set,
(A – B) ∪ (B – A) and it is denoted by A △ B.
A △ B = (A – B) ∪ (B – A) = {x : x ∈ A,x ∈ B,x ∉ A ∩ B}

- Properties of the operation of union
- Commutative law:
A ∪ B = B ∪ A
- Associative law:
(A ∪ B) ∪ C = A ∪ (B ∪ C)
- Law of identity:
A ∪ φ = A
- Idempotent law:
A ∪ A = A
- Law of U:
U ∪ A = U
- Properties of operation of intersection
- Commutative law:
A ∩ B = B ∩ A
- Associative law:
(A ∩ B) ∩ C = A ∩ (B ∩ C)
- Law of φ and U:
φ ∩ A = φ and U ∩ A = U
- Idempotent law:
A ∩ A = A
- Distributive law:
A ∩ (B ∩ C) = (A ∩ B) ∪ (A ∩ C)
- Properties of complement of sets
- Complement laws:
A ∪ A’ = U
A ∩ A’ = φ
- De-Morgan’s law:
(A ∪ B)’ = A’ ∩ B’
(A ∩ B)’ = A’ ∪ B’
- Law of double complementation:
(A’)’ = A
- Laws of empty set and universal set:
φ’ = U and U’ = φ
- Operations on sets
- A – B = A ∩ B'
- B – A = A' ∩ B
- A – B = A ⇔ A ∩ B = φ
- (A – B) ∪ B = A ∪ B
- (A – B) ∩ B = φ
- A ⊆ B ⇔ B' ⊆ A'
- (A – B) ∪ (B – A) = (A ∪ B) – (A ∪ B)
- Some more important results:
Let A, B and C be three sets. Hence,
- A – (B ∩ C) = (A – B) ∪ (A – C)
- A – (B ∪ C) = (A – B) ∩ (A – C)
- A ∩ (B – C) = (A ∩ B) – (A ∩ C)
- A ∩ (B △ C) = (A – B) △ (A – C)
- Counting Theorems
- If A and B are finite sets, then the number of elements in the union of two sets is given by
n(A ∪ B) = n(A) + n(B) – n(A ∪ B)
- If A and B are finite sets and A ∩ B = φ, then
n(A ∪ B ) = n(A) + n(B)
- n(A ∪ B) = n(A – B) + n(B – A) + n(A ∩ B)
- n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(B ∩ C) – n(A ∩ B) - n(A ∩ C) + n(A ∩ B ∩ C)
- Number of elements in exactly two of the sets =
n(A ∩ B) + n(B ∩ C) + n(C ∩ A) – 3n(A ∩ B ∩ C)
- Number of elements in exactly one of the sets =
n(A) + n(B) + n(C) – 2n(A ∩ B) – 2n(B ∩ C) – 2n(C ∩ A) + 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)
- Number of elements in the power set of a set with n elements = 2n
Number of proper subsets in the power set = 2n – 1