Definition and Representation A set is a well-defined collection of objects; The objects are called...
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Transcript of Definition and Representation A set is a well-defined collection of objects; The objects are called...
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:
EE5522
113344
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
????
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.