# 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**A**, is the set of all elements of U which are not in set A.^{c} - 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

- Commutative law:
**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)

- Commutative law:
**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

- Complement laws:
**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)

- If A and B are finite sets, then the number of elements in the union of two sets is given by
- Number of elements in the power set of a set with n elements = 2
^{n}

Number of proper subsets in the power set = 2^{n}– 1