# Sets

**Definition**

A **set** is a collection of explicitly-defined distinct *elements*.

## Elements of a Set

If 2, 4, 6, 8 are elements of *A*, we can then write

or it can be written using the **set-builder** notation

read as "*A* is the set of all *x* such that *x* is an even digit".

The phrase “2 is an element of *A*” is equivalent to the statement “2 belongs to *A*”. In symbol, we write

It follows that

to indicate that “5 is not an element of *A*”.

## Universal Set, *U*

*Universal set* consists all elements under consideration.

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## Null Set

An *empty set*, or a set with no element, is called a null set and is denoted by $\varnothing$.

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## Subset

If all elements of set *A* belongs to set *B*, then *A* is a *subset* of *B*. In symbol we write

to respectively indicate that "*A* is contained in *B*" or "*B* contains *A*".

The negation of this is

**Example**

*A*= {

*x*:

*x*is an even digit} and

*B*= {

*x*:

*x*≥ 0,

*x*is integer} then

*A*⊆

*B*or {0, 2, 4, 6, 8} ⊆ {0, 1, 2, 3, …}

**Note:** All sets under consideration belongs to a particular universal set.

## Complement of a Set

All set elements which belongs to *U* but do not belong to *A* is called the *complement* of set *A* and is denoted by *A ^{c}*.

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## Disjoint Sets

Two sets are said to be *disjoint* if the sets contains no common element.

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## Union

The *union* of two sets is the set of all elements which belongs to either or both sets.

The union of *A* and *B* is denoted by *A* ⋃ *B*.

**Example**

*A*= {1, 2, 3, 4, 5} and

*B*= {4, 5, 6, 7, 8} then

*A*⋃

*B*= {1, 2, 3, 4, 5, 6, 7, 8}

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## Intersection

The *intersection* of two sets is the set of all elements which belongs to both sets.

The intersection of *A* and *B* is denoted by *A* ⋂ *B*.

**Example**

*A*= {1, 2, 3, 4, 5} and

*B*= {4, 5, 6, 7, 8} then

*A*⋂

*B*= {4, 5}

**Note:** If *A* and *B* are disjoint sets, then *A* ⋂ *B* = $\varnothing$.

## Difference

The *difference* of two sets, also called **relative complement**, is the set of elements which belongs to one set but do not belong to the other set.

The difference of *A* and *B* is denoted by *A*\*B*.

**Example**

*A*= {1, 2, 3, 4, 5} and

*B*= {4, 5, 6, 7, 8} then

*A*\

*B*= {1, 2, 3}

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## Symmetric Difference

The *symmetric difference* of two sets consists of those elements which belongs to either of the two sets but excludes the elements that belongs to both of the two.

The symmetric difference of *A* and *B* is denoted by *A* ⊕ *B*.

**Example**

*A*= {1, 2, 3, 4, 5} and

*B*= {4, 5, 6, 7, 8} then

*A*⊕

*B*= {1, 2, 3, 6, 7, 8}

**Note:** If *A* and *B* are disjoint sets, then *A* ⊕ *B* = *A* ⋃ *B*.