## OPERATIONS ON SETS:

### Union:

Let X and Y be two sets. We define the union of these sets as follows

X∪Y= { z/z∈X or z∈Y}

Here we see that X∪Y contains all the elements of X and all elements of Y.

#### Venn Diagram for Union of sets:

**Example:**

Let A={ a,b,c,d} and B={ a,c,e}. Then A∪B is given by

A∪B={a,b,c,d,e}

### Intersection:

Let X and Y be two sets. We define the intersection of these sets as follows

X∩Y={z/z∈X and z∈Y}

Here we see that X∩Y contains only the elements that are contained in both X and Y.

Note: X∩Y ⊆X and also X∩Y ⊆Y

#### Venn diagram for Intersection of sets:

**Example:**

Let A={1,2,3,4,5} and B={3,4,5,6}.Then A∩B is given by

A∩B={3,4,5}

### Set Difference:

Let X and Y be two sets. We define the set difference as follows

X\Y={z/z∈X but z∉Y}

Here we see that X\Y contains only elements of X but does not contain the elements of Y.

Some authors use X-Y to denote set difference but most commonly used notation for set difference is X\Y.

#### Venn diagram for set difference:

**Example:**

Let A={5,10,15,20} and B={6,10,12,18,24}.Then A\B is given by

A\B={5,15,20}

### Symmetric difference:

Let X and Y be two sets. we define the symmetric difference of these sets

x ΔY=(X\Y) ∪ (Y\X).

XΔY contains all elements in X∪Y but does not contain the elements in X∩Y.

#### Venn diagram for the symmetric difference:

**Example:**

Let A={ x\x is a positive integer less than 12} i.e. A={1,2,3,4,5,6,7,8,9,10,11} and B={ 1,2,4,6,7,8,12,15}

Then AΔB is given by

AΔB={3,5,9,10,11,12,15}

### Complement:

Let U be the universal set. and let X be a subset of U i.e. X⊆U then U\X is called the complement of the set X with respect to U. We denote U\X by X’ and is called the complement of X.

The set difference X\Y can be viewed as the complement of Y with respect to X.

#### Venn Diagram for complement of set:

**Example:**

Let U={ 1,2,3,4,5,6,7,8,9,10} and A={2,4,6,8,10} Then A’ is given by

A’={ 1,3,5,7,9}.

### Disjoint Sets:

Let X and Y be two sets. These two sets X and Y are said to be disjoint if they does not have elements in common. i.e. X∩Y={}

#### Venn Diagram for Disjoint sets:

**Example:**

Let A={a,b,c,d} and B={p,q,r}

Here we see that A and B don’t have elements in common. thus A and B are disjoint sets.