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K.V.S BEAWAR (Raj)

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class : 11 Science name : Khushwant SETS Mathematics project

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contents History of sets Sets Sets representation Types of set History of Venn Union of sets Intersection of set Complements of set

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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 .

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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.

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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.

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sets representation There are two ways to represent sets : Roster or tabular form . Set-builder form .

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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 }

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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 }

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examples of sets in maths

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types of sets Empty set. Finite & Infinite sets. Equal sets. Subset. Power set. Universal set.

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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 }

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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)

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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

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subsets

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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 } }

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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]

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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.

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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 .

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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.

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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 }

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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).

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Venn consist of rectangles & closed cure usually circles. The universal set is represented usually by rectangle & its subsets by circle.

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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.

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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.

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some properties of union

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intersection of sets : The intersection of two sets A & B is the set of all those elements which belong to both A & B.

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some properties of intersection :

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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.

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properties of complements of sets :

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Laws of double complementation : ( A’ )’ = A Laws of empty set and universal set : f’ = U & U’ = f THE END