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HISTORY OF SETS. The theory of sets was developed by German mathematician Georg Cantor. He first encountered sets while working on “problem on trigonometric series”. SETS are being used in mathematics problem since they were discovered..

SETS. Collection of object of a particular kind, such as, a pack of cards, a crowd of people, a cricket team etc. In mathematics of natural number, prime numbers etc..

A set is a well defined collection of objects .. Elements of a set are synonymous terms. Sets are usually denoted by capital letters. Elements of a set are represented by small letters..

SETS REPRESENTATION. Roster or tabular form Set-builder form.

ROSTER OR TABULAR FORM. In roster form, all the elements of set are being separated by commas (,)and are enclosed within brackets. E.G. : sets of 1,2,3,4,5,6,7,8,9,10.

SET-BUILDER FORM. In set-builder form, all the elements of a set possess a single common property which is not possessed by an element outside the set. E.G. : set of natural numbers k K =.

EXAMPLE OF SETS IN MATHS. N : THE SET OF ALL NATURAL NUMBERS Z : THE SET OF ALL INTEGERS Q : THE SET OF ALL RATIONAL NUMBERS R : THE SET OF ALL REAL NUMBERS Z+ : THE SET OF ALL POSITIVE INTEGERS Q+ : THE SET OF POSITIVE RATIONAL NUMBERS R+ : THE SET OF POSITIVE REAL NUMBERS.

TYPES OF SETS. EMPTY SETS FINITE AND INFINITE SETS EQUAL SETS SUBSET POWER SET UNIVERSAL SET.

THE EMPTY SET. A SET WHICH DOESN’ T CONTAINS ANY ELEMENT IS CALLED THE EMPTY SET OR NULL SET OR VOID SET, DENOTED BY SYMBOL OR Ø. E.G. : LET R =.

FINITE AND INFINITE SETS. A SET WHICH IS EMPTY OR CONSISTS OF A DEFINITE NUMBERS OF ELEMENTS IS CALLED FINITE OTHERWISE, THE SET IS CALLED INFINITE . E.G. : LET R BE SET OF A POINTS ON A LINE THEN R IS INFINITE . LET K BE A SET OF DAYS OF WEEKS THEN K IS FINITE..

EQUAL SETS. GIVEN TWO SETS K AND R ARE SET TO BE EQUAL IF THEY HAVE EXACTLY THE SAME ELEMENT ANDWE WRITE K = R OTHERWISE THE SET ARE SAID TO BE UNEQUAL AND WE WRITE K = R . E.G.: LET K = AND R = THEN K = R ..

SUBSETS. A SET R IS SAID TO BE A SUBSET OF A SET K IF EVERY ELEMENTS OF R IS ALSO A ELEMENT OF K . R ⊆ K THIS MEANS ALL THE ELEMENTS OF R CONTAINED IN K ..

POWER SET. THE SET OF ALL SUBSET OF A GIVEN SET IS CALLED POWER SET OF THAT SET. THE COLLECTION OF ALL SUBSET OF A SET K IS CALLED THE POWER SET OF DENOTED BY P(K).IN P(K) EVERY ELEMENT IS A SET. IF K = [1,2} P(K) =,,}.

UNIVERSAL SETS. Universal set is set which contains all object, including itself . e.g. : the set of real number would be the universal set of all other sets of number NOTE : excluding negative root.

A Venn diagram or set diagram is a diagram that shows all possible logical relations between à finite collection of sets. Venn diagrams were conceived around 1880 by John Venn. They are used to each elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics linguistics and computer science..

VENN CONSIST OF RECTANGLES AND CLOSED CURVES USUALLY CIRCLES. THE UNIVERSAL IS REPRESENTED USUALLY BY RECTANGLES AND ITS SUBSETS BY CIRCLE ..

ILLUSTRATION 1. IN FIG U=(1, 2, 3, ..... 10] IS THE UNIVERSAL SET OF WHICH A=(2, 4, 3, .….10) IS A SUBSET..

ILLUSTRATION 2. IN FIG U = [1, 2, 3, ...., 10] IS THE UNIVERSAL SET OF WHICH A=(2, 4, 6, 8, 10] AND B =[4, 6] ARE SUBSETS, AND ALSO B C A.

OPERATIONS ON SETS. UNION OF SETS: the union of two sets A and B is the set C which consist of all those element which are either in A or B or in both ..

SOME PROPERTIES OF THE OPERATION OF UNION. AUB=BUA (commutative law) 2) (AUB) UC=AU (BUC) ( associative law) 3) AUØ = A (law of identity Element ) 4)AUA = A ( IDEMPOTENT LAW) 5 ) UUA = A ( law of U ).

SOME PROPERTIES OF THE OPERATION OF INTERSECTION.

COMPLIMENT OF SETS. Let U=(1, 2, 3,] now the set of all those element of U which doesn't belongs to A will be called as A compliment ..

PROPERTIES OF COMPLEMENT OF SETS. Complement laws: 1 ) AUA ' = U 2)A Ո A ' = Ø 2) De Morgan's law: 1 ) (AUB)' = A' Ո B‘ 2 ) ( A Ո B ) = A'UB ' 3)Laws of double complementation : (A')' = A 4) Laws of empty set and universal set: ع = U & U' = Ø.

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