Slide 1: K.V.S BEAWAR (Raj)
Slide 2: class : 11 Science
name : Khushwant SETS Mathematics project
Slide 3: contents History of sets
Sets
Sets representation
Types of set
History of Venn
Union of sets
Intersection of set
Complements of set
Slide 4: HISTORY OF SETS The theory of sets was developed by
German mathematician Georg Cantor
(1845-1918) . He first encountered sets while working on “ Problems on
Trigonometric series”. SETS are being
Used in solving mathematics problems since they were discovered .
Slide 5: sets Collection of object of a particular
kind, such as, a pack of cards, a
crowed of peoples, a cricket team, etc, In mathematics of natural no.,
points, prime no., etc.
A set is a well defined collection of
object.
Slide 6: Elements of a set are synonymous terms.
Sets are usually denoted by capital letters.
Elements of a set are represented by small
letters. a set is a well defined collection
of object.
Slide 7: sets representation There are two ways to represent sets :
Roster or tabular form .
Set-builder form .
Slide 8: set-builder form In set-builder form, all the elements of a set possess a single common property which is not possessed by any element
outside the set .
e.g. :
set of natural numbers k .
k= { x : x is a natural no }
Slide 9: roster form In roster form all the elements of sets are listed, the elements are being
separated by commas & are enclosed
within braces { } .
e.g. :
set of 1,2,3,4,5,6,7,8,9,10 .
{ 1,,2,3,4,5,6,7,8,9,10 }
Slide 10: examples of sets in maths
Slide 11: types of sets Empty set.
Finite & Infinite sets.
Equal sets.
Subset.
Power set.
Universal set.
Slide 12: the empty set A set which doesn’t contains any element
is called the empty set or null set or void set, donated by symbol f or { } .
e.g. : let R = { x : 1 < x < 2, x is a natural
number }
Slide 13: finite & infinite sets A set which is, empty or consist of a definite no. of elements is called finite otherwise, the set called infinite .
e.g. : let k be the set of the days of the week .
Then k is finite. (finite)
let r be the set of points on a line.
Then R is infinite. (infinite)
Slide 14: equal sets Two sets k & R are said to be equal if they have exactly the same elements and we write k=R . Otherwise, the sets are said to be unequal and we write k?R.
e.g. :
let k = { 1,2,3,4,} & R= { 1,2,3,4 }.
then k=R
Slide 15: subsets
Slide 16: power set The set of all the subsets of a given set is called power set of that set.
The collection of all subsets of a set k
is called the power set of k denoted by
P ( k ) . In P ( k ) every element is a set.
if k = { 1,2 }
P ( k ) = { f , { 1 } , { 2 } , { 1 , 2 } }
Slide 17: universal set The super set of all the given type of sets would be called as universal set
of all the other given type of sets.
e.g. : the set of real numbers would be the
universal set of all the other sets
of numbers.
Note : [excluding negative roots]
Slide 18: Subsets of r The set of natural no. N={ 1,2,3, …}
The set of integers Z={… , -2,-1,0,1,2,…}
The set of rational no. Q={ x : x = p/q ,
p,q are integers and q ? 0 }
Note : members of Q also include negative
numbers.
Slide 19: Intervals of subsets of r The interval denoted as ( a , b ) , a & b are Real numbers ; is an open interval , means including all the elements between a to b but excluding a & b .
Slide 20: The interval donated as [ a , b ] , a & b are Real numbers ; is an closed interval , means including all the elements between a to b &including a & b.
Slide 21: types of intervals ( a , b ) = { x : a < x < b }
[ a , b ] = { x : a = x = b }
[ a , b ) = { x : a = x < b }
( a , b ] = { x : a < x = b }
Slide 22: history of Venn diagrams Most of the relationships of sets can be represented using Venn diagrams . Venn are named after the English logician, Johan Venn
(1834-1883).
Slide 23: Venn consist of rectangles & closed cure usually circles. The universal set is represented usually by rectangle & its subsets by circle.
Slide 24: illustration 1. In fig 1., U =
{ 1, 2, 3, …, 10 } is the universal set of which
A = { 2, 4, 6, 8, 10 } is a subset.
Slide 26: Union of sets : The union of two sets A & B is the set C which consists of all those elements which are either in A or B or in both.
Slide 27: some properties of union
Slide 28: intersection of sets : The
intersection of two sets A & B is the set of all those elements which belong to both A & B.
Slide 29: some properties of intersection :
Slide 30: complements of sets : Let U = { 1, 2, 3, 4, …, 10 } & A = { 1, 2, 3 }
Now the set of all those elements of U
which doesn‘t belongs to A will be called
as A’ or A complement.
Slide 31: properties of complements of sets :
Slide 32: Laws of double complementation :
( A’ )’ = A
Laws of empty set and universal set :
f’ = U & U’ = f THE END