Definition and Representation A set is a well-defined collection of objects;

The objects are called elements or members of the set;

A set can be represented by; (a) A description in words:

A = (the first 5 natural numbers)

(b) listing all its elements or members:

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

(c) Set –builder notation:

A = { x: 1 ≤ x ≤ 5, x R }

A set can also be represented diagrammatically by means of a Venn diagram:

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RR

An element of a set is denoted by the symbol 4 {2 ,4, 6, 8, 10 }

A non–element of a set is denoted by the symbol

7 { 1, 2, 3 ,4, 5 } The number of elements in a set A is

written as n (A)A = {1, 2, 3, 4, 5}

Hence n( A ) = 5

Elements of a set

Two sets are said to be equal or identical if

both sets have exactly the same elements

which may not be shown in the same order.e.g. { a, b, c, d } = { c, d, b, a }

Equal Sets

Empty Set

An empty or null set has no element. It is denoted by the symbol Ø or { }

e.g. (a) { dogs with 3 pairs of legs } = Ø

e.g. (b) { months with 29 days in 2003 } = Ø

A universal set contains all the elements

It is denoted by the letter ‘ ‘

e.g. (a) The universal set of integers

= {…,-2, -1, 0, 1, 2,…}

(b) The universal set of animals

= { hens, dogs, lions, cats,…}

Universal Set

Finite and Infinite sets

A finite set has a limited number of elements:

e.g. { even numbers between 11 and 20 }

is equal to { 12, 14, 16, 18 }

An infinite set has an unlimited numbers of elements:

e.g. { odd numbers } = { 1, 3, 5, 7, 9,…}

The complement of a set A is the set of those members in the universal set that are not members of A.

The complement of the set is denoted by A’.

e.g. Given that = {1, 2, 3, 4, 5, 6, 7, 8, 9}

and A = {1, 5, 8, 9 } list the numbers of the set A’. Illustrate the set A’ by

means of a Venn diagram.

Complement Set

Solution:

= {1, 2, 3, 4, 5, 6, 7, 8, 9 }

A = { 1, 5, 8, 9, }

Hence A’ = { 2, 3, 4, 6, 7 }

1, 5, 8, 9

AA

2, 3, 4, 6, 7

If two sets A and B have no common element, they are said to be disjoint.

e.g. If A = { even numbers }

and B = { odd numbers }

then A and B are disjoint

Disjoint Set

Subset

The set A is a subset of the set B if every element of A is an element of B.

Subset is denoted by the symbol :

A = { 3, 5 } B = { 1, 3, 5, 7, 9, 11} A B

A subset can be illustrated by means of a Venn diagram:

Every set is a subset of itself:{a, b, c, } { a, b, c}

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Intersection

Intersection

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Intersection

n(AB)

A B

Union

Union

n(A B) = n(A) + n(B) – n(AB)

A B

History

George Cantor, (1845-1918), German mathematician – founder of “The Set Theory”

Set the foundations of many advanced mathematical works.

Concepts of sets can help in classifying & counting things.

Make learning of Mathematics more meaningful

Real Life Examples

A set is a collection of things.• Some examples:• A coin collection • The English alphabet • Even numbers • Odd numbers

Hands-on Activities

Get students to sort items by colour, size, and shape. • Some examples • Things to sort by colour: crayons, markers, backpacks,

clothes• Things to sort by shape: tables, block• Things to sort by size: books, shoes, students

Buttons

Divide the class into groups of three or four students each.

Give each group a small container of assorted buttons.

Ask each student to sort the buttons according to colours, shape & size.