Sets {Maths}
- 1. BY – PRIYAL DAFTARI
- 2. SET SET IS A WELL-DEFINED COLLECTION OF OBJECTS.WELL-DEFINED MEANS THAT THERE IS A DEFINITE CRITERIA TO DETERMINE WHETER AN OBJECT BELONGS TO THE GIVEN SET OR NOT. COLLECTION OF EVEN NUMBER TILL 10 E={2,4,6,8,1,0} ELEMENTS OF SETS THE OBJECTS IN THE SETS ARE CALLED ELEMENTS OR MEMBERS OF THE SETS .WE PUT THE ELEMENTS OF A SET WITHIN CURLY BRACES {} AND USUALLY DENOTE A SET BY A CAPITAL LETTER. ODD NUMBERS TILL 5 O={1,3,5} HERE 1,3 AND ARE THE ELEMENTS OF SET O. REPRESENTATION OF SETS THERE ARE 2 WAYS TO REPRESENT A SET ROSTER FORM-THE METHOD IN WHICH WE LIST THE ELEMENTS OF A GIVEN SET IS CALLED THE ROSTER FORM OR THE LISTING METHOD. MULTIPLES OF 2 TILL 14 M={2,4,6,8,10,12,14}-----ROSTER FORM SET-BUILDER FORM-THE METHOD IN WHICH WE LIST THE PROPERTY OR PROPERTIES SATISFIED BY THE ELEMENTS OF THE SET IS KNOWN AS THE SET-BUILDER OR THE RULE METHOD. COLLECTION OF ALL THE INTEGERS BETWEEN -10 AND -20 S={x/x is an integer,-20<x<-10}-------SET-BUILDER FORM CARDINAL NUMBER OF A SET CARDINAL NUMBER OF SET A IS THE NUMBER OF DISTNCT ELEMENTS N THE SET.IT IS DENOTED AS = n(A) A={2,3,4,5} CARDINAL NUMBER OF SET A= n(A)=4
- 3. TYPES OF SETS 1.EQUIVALENT SETS IF THE NUMBER OF ELEMENTS INTWO SETS ARE EQUAL,THEN THEY ARECALLED EQUIVALENT SETS. P={N,M,O,P,Q} AND Q={A,B,C,D,E} n(P)=5 and n(Q)=5 SET P AND SET Q ARE EQUIVALENT. 2.EQUAL SETS TWO SETS ARESAIDTO BEEQUAL IF THEYHAVE THESAMEELEMENTS. A={5,7,9} AND B={9,5,7} THEHAVE THESAMEELEMNTS AND THUSARE EQUAL SETS. 3.FINITESETS A SET CONTAINSONLY A FINITENUMBER OF ELEMENTS,THEN ITIS CALLED A FINITESET. COLLECTION OF MONTHSIN A YEAR M={JANUARY,FEBRUARY,MARCH,APRIL,MAY,JUNE,JULY,AUGUST,SEPTEMBER,OCTOBER,NOVEMBER,DECEMBER} THE ELEMENTS OF THE SET ARE LIMITEDAND THUSTHISSET IFSA FINITESET. 4.INFINITESETS THENUMBER OF ELEMENTS OF A SET CANNOT BE COUNTED,THEN IT ISAN INFINITESET. COLLECTION OF WHOLENUMBERS W={0,1,2,3,4,5,6…..} IN THISSET THEELEMENTS ARE UNLIMITEDAND INFINITE THUSTHISSET ISAN INFINITESET. 5.SINGLETON SETS A SET THATCONTAINS ONLY ONE ELEMENTS IS KNOWN ASA SINGLETON SET. COLLECTION OF EVEN PRIME NUMBERS E={2}----SINCE THISSET ONLY HASA SINGLE ELEMENT IT ISCALLED A SINGLETON SET. 6.EMPTY SET A SET THATCONTAINS NO ELEMENTS IS KNOWN AS AN EMPTY SET. COLLECTION OF A NEGATIVE WHOLE NUMBER WN={ }-----SINCETHERE IS NO NEGATIVE WHOLENUMBER THEREISNO ELEMENT ANDTHUSTHESET ISAN EMPTY SET.
- 4. 1. DISJOINT SETS TWOSETS HAVINGNO COMMON ELEMENTSINCOMMON AREKNOWNASDISJOINT SETS E={2,4,6}ANDO={1,3,5}—SINCETHESETSHAVE NTHNGINCOMMON THEY AREKNOWN ASDISJOINTSETS. 2.OVERLAPPING SETS TWOSETSHAVINGATLEAST ONE ELEMENT INCOMMON AREKNOWN ASOVERLAPPIN SETSOR INTERSECTING SETS. A={5,6,7,8,9,}AND B={3,6,9} SINCETHETWOSETSHAVE22ELEMENTSIN COMMON THEY AR SAID TOBE INTERSECTINGSETS. 3.UNIVERSAL SETS THESETTHATCONTAINSALL THEELEMENTSOF THESETSUNDERCONSIDERATIONISKNOWN ASTHE UNIVERSALSET.ITISDENOTED USINGTHESYMBOL U SET A ISA SETOF WHOLE NUMBERSTILL8 ANDSETB ISA SETOF NATURALNUMBERSTILL10 A={O,1,2,3,4,5,6,7,8}ANDB={1,2,3,4,5,6,7,8,9,10} U={0,1,2,3,4,5,6,7,8,910}