# Set

### Set

###### Set

A set is a collection of well-defined objects

A set is well-defined if we are able to tell whether or not any particular object is an element of the set

Example:

Which of the following are well-defined sets?

(1)   All the tall girls of the school

(2)   All the letters in the word "mathematics"

(3)   All the hardworking teachers in a school

(4)   All the honest members in the family

All the letters in the word "mathematics" is well-defined sets

B = {m, a, t, h, e, i, c, s}

The following conventions are used with sets:

• Capital letters are used to denote sets
• Curly braces { } denote a list of elements in a set
• Lowercase letters are used to denote elements of sets

Example:

A is the set of vowels

A = {a, i, u, e, o}

The elements in the sets are depicted in either the statement form, set-builder notation form or roster form

1. Statement form

A = {prime numbers between 10 and 30}

2. Set-builder notation

A = {x| 10 < x < 30, x ∈ prime numbers}

We read it as, "A is the set of all x such that x is more than10 but less than 30 and x belongs to prime numbers"

3. Roster form

A = {11, 13, 17, 19, 23, 29}

##### Elements Of a Set

An element is a member of a set.

Notation:

∈  means “is an element of”

∉  means “is not element of”

##### Cardinality

The number of elements of a set is expressed as n(A)

Example:

A = {positive integers between −3 and 3}

A = {1, 2}

1 ∈ A  {1 is an element of A}

2 ∈ A  {2 is an element of A}

3 ∉ A {3 is not element of A}

n(A) = 2

##### Empty Set and Infinite Set

Null or empty set

Set with no elements

Example:

The sets of triangles having 4 sides

It is an empty set, because every triangle has 3 sides

Infinite set

The element cannot be listed (unlimited)

Example:

• Set of all point in a line segment
• Set of all natural numbers
• Set of all integers

##### Subset

$$\text{A}$$ is a subset of $$\text{A}$$ ($$\text{A}\subseteq \text{B}$$) if and only if every element of $$\text{A}$$ is in $$\text{B}$$.

The total number of subsets of a set with n elements is $$2^{n}$$.

Example:

$$\text{P} = \lbrace 5, 10, 15, 20 \rbrace$$

$$\text{Q} = \lbrace 5, 10, 15, 20, 25, 30 \rbrace$$

$$\text{P}$$ is a proper subset of $$\text{Q}$$ because the element 25 and 30 is not in the set $$\text{P}$$. ($$\text{P} \subset \text{Q}$$)

The number of subset of $$\text{Q}$$ is $$2^6 = 64$$.

Example:

Find the number of subsets of  $$\text{P} = \lbrace \text{a, b, c, d, e, f}\rbrace$$  that contain exactly 3 elements

The total number of subsets of $$\text{P}$$ is $$2^{6}$$

Using Pascal's triangle:

So, the number of subsets of $$\text{P}$$ that contain exactly 3 elements is 20.

##### Equal Sets and Equivalent Sets

Two sets A and B are equal if they contain exactly the same elements

Example:

A = {1, 2, 4} and B = {4, 2, 1}

So set A is equal set B (A = B)

Two sets A and B are said to be equivalent if they have the same cardinality i.e. n(A) = n(B)

Example:

A = {a, i, u, e, o} and B = {2, 3, 5, 7, 11}

Set A is equivalent set B (A ∼ B) because n(A) = n(B) = 5

##### Universal Set

A universal set is the set of all elements or members of all related sets

Example :

Describe one possible universal set for each of the following sets!

1. {4, 6, 8, 10}

2. {Neptune, Saturn}

3. {cat, dog}

Solution:

1. U = the set of even numbers

2. U = the set of planets in our solar system

3. U = the set of pets

##### Disjoint Set and Joint Set

Disjoint set is two or more sets which have no elements in common

Example:

$$\text{U} = \lbrace 1, 2, 3, 4, 6, 7, 8, 9\rbrace$$

$$\text{A} = \lbrace 1, 2, 3, 4\rbrace$$

$$\text{B} = \lbrace 6, 7, 8, 9\rbrace$$

Set A and B are called disjoint set

Joint set is two or more sets which have elements in common

Example:

$$\text{U} = \lbrace 1, 2, 3, 5, 7, 9, 11 \rbrace$$

$$\text{A} = \lbrace 1, 3, 5, 7, 9, 11\rbrace$$

$$\text{B} = \lbrace 2, 3, 5, 7\rbrace$$

Set A and B is joint set because they have elements in common {3, 5, 7}