Table of Contents

### Union of Sets

The union of two sets A & B is the set consisting of all the elements which belong to either A or B or both.

The union of set A & B is denoted by A U B , read as A union B.

Symbolically, **A U B = {x / x ∈ A or x ∈ B or x ∈ both A & B}**

Venn diagram for Union of Set A & B is drawn below. A U B is indicated by blue color.

#### Properties of Union of Sets

- A ⊂ (A U B) and B ⊂ (A U B)
- A U Φ = A, for every set A
- A U A = A, for every set A
- A U B = B U A (Commutative Property)
- (A U B) U C = A U (B U C) (Associative Property)
- If B ⊂ A then A U B = A & if A ⊂ B then A U B = B
- A U B = Φ
- A U (B ∩ C) = (A U B) ∩ (A U C) (Distributive Law of Union over Intersection)
- If A ⊂ B & C ⊂ D then (A U B) ⊂ (B U D)

### Intersection of Sets

The intersection of two sets A & B is the set consisting of all elements which belong to both the set A & B

The intersection of A & B is denoted by A ∩ B, read as A intersection B.

Symbolically, **A ∩ B = {x / x ∈ A & x ∈ B}**

Venn diagram for Intersection of Set A & B is drawn below. A ∩ B is indicated by blue color.

#### Properties of Intersection of Sets

- (A ∩ B) ⊂ A and (A ∩ B) ⊂ B
- A ∩ Φ = Φ, for every set A
- A ∩ A = A, for every set A
- A ∩ B = B ∩ A (Commutative Property)
- (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative Property)
- If B ⊂ A then A ∩ B = B & if A ⊂ B then A ∩ B = A
- If A ⊂ B & B ⊂ C then A ⊂ (B ∩ C)
- A ∩ (B U C) = (A ∩ B) U (A ∩ C) (Distributive Law of Intersection over Union)
- If A ⊂ B & C ⊂ D then (A ∩ B) ⊂ (B ∩ D)

#### Examples of Union and Intersection

### Complement of a Set

The complement of any set is the set of all those elements which do not belong to that set itself.

If U is the universal set & A ∈ P(U), then complement of set A is the set U – A & is denoted by A’.

Symbolically, **A’ = {x / x ∈ U, x ∉ A}**

Venn diagram for Complement of Set A is drawn below. A’ is indicated by purple color.

#### Properties of Complement of Set

- A ∩ A’ = Φ
- A U A’ = U (Universal set)
- U’ = Φ
- Φ’ = U
- (A’)’ = A
- If A ⊂ B then B’ ⊂ A’
- (A ∩ B) U (A ∩ B’)=A
- (A ∩ B)’ = A’ U B’ (De Morgan’s Law)
- (A U B)’ = A’ ∩ B’ (De Morgan’s Law)

#### Examples of Complement of Sets

### Difference of Two Sets

The difference of two sets A & B is the set of all those elements which belong to set A but not set B. It is denoted by A – B & read as A difference B.

Symbolically, **A – B = {x / x ∈ A but x ∉ B}**

Similarly, the difference of two sets B & A is denoted by B – A.

Symbolically, **B – A = {x / x ∈ B but x ∉ A}**

The Difference of two sets can be shown below in shaded region by Venn diagram.

#### Properties of Difference of Sets

- A – B ⊂ A & B – A ⊂ B
- A – B, A ∩ B, B – A are mutually disjoint sets.
- A – (A – B) = A ∩ B & B – (B – A) = A ∩ B
- If A = B =>(A – B) = Φ
- A – B = A ∩ B’
- A – Φ = A
- A ⊂ B =>A – B = Φ
- U – A = A’
- A U B = (A – B) U B

### Symmetric Difference Set

A difference set is called a Symmetric Difference of two sets if it contains all those elements which are in set A, but not in set B or those which are in set B but not in set A.

It is the union of two different set A & B. It is denoted by A Δ B = (A – B) U (B – A)

Symbolically, **A Δ B = {x / x ∈ A or x ∈ B but x ∉ A & B}**

The Symmetric difference of two sets can be shown below in shaded region by Venn diagram.

#### Properties of Symmetric Difference Sets

- A Δ B = B Δ A
- A Δ A = Φ
- A Δ (A ∩ B) = A – B
- (A Δ B) U (A ∩ B) = A U B
- A Δ B = (A U B) – (A ∩ B)

### Order Pair

An order pair of two elements x & y written in bracket (x, y) such that one element, say x is designated as first member & y as the second member. For example :

- The natural number & their squares can be written by order pair, (1, 1), (2, 4), (3, 9), (4, 16)…
- The point on 2-dimension graph can be represented by an order pair (x, y), where x is the first co-ordinate of x-axis & y is the second co-ordinate of y-axis.
- Two order pairs (p, q) & (x, y) will be equal if and only if, p = x & q = y.

### Cartesian Product

If A & B be any two non empty sets, then the set of all ordered pair whose first element belong to set A & second element belong to set B is called the Cartesian Product of A & B in that order. It is denoted by A X B.

Symbolically, **A X B = {(x, y) : x ∈ A and y ∈ B}**

#### Properties of Cartesian Product

- Though the number of elements in set A & B are same, A X B ≠ B X A
- A X B = B X A if and only if A = B
- If the number of elements in set A is x & in set B is y then the number of elements in product set A X B are xy.