Sets and Operations on Sets - PDF
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This document explains sets, their various types, methods for writing sets (roster and rule method), and operations, such as union, intersection, complement, difference, and symmetric difference. It features diagrams illustrating set theory concepts.
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# SETS ## TERMS OF SET - A SET IS AN UNODERED WELL-DEFINED COLLECTION OF OBJECTS. - THE OBJECTS ARE CALLED ELEMENTS OR MEMBERS OF THE SET. - A FINITE SET IS A SET WHOSE ELEMENTS ARE LIMITED OR COUNTABLE, AND THE LAST ELEMENT CAN BE IDENTIFIED. - AN INFINITE SET IS A SET WHOSE ELEMENTS ARE UNLIMITE...
# SETS ## TERMS OF SET - A SET IS AN UNODERED WELL-DEFINED COLLECTION OF OBJECTS. - THE OBJECTS ARE CALLED ELEMENTS OR MEMBERS OF THE SET. - A FINITE SET IS A SET WHOSE ELEMENTS ARE LIMITED OR COUNTABLE, AND THE LAST ELEMENT CAN BE IDENTIFIED. - AN INFINITE SET IS A SET WHOSE ELEMENTS ARE UNLIMITED OR UNCOUNTABLE, AND THE LAST ELEMENT CANNOT BE SPECIFIED. - A UNIT SET IS A SET WITH ONLY ONE ELEMENT, ALSO A SINGLETON. - A UNIQUE SET WITH NO ELEMENTS IS CALLED EMPTY SET, OR NULL SET. - ALL SETS UNDER INVESTIGATION IN ANY APPLICATION OF SET THEORY ARE ASSUMED TO BE CONTAINED IN SOME LARGELY FIXED SET CALLED THE UNIVERSAL SET, DENOTED BY U. - THE CARDINALITY OR CARDINAL NUMBER OF A SET IS THE NUMBER OF ELEMENTS OR MEMBERS IN THE SET, THE CARDINALITY OF SET A IS DENOTED BY N(A). ## METHODS OF WRITING ### ROSTER METHOD - THE ELEMENTS OF THE SET ARE ENUMERATED AND SEPARATED BY A COMMA. ALSO CALLED TABULATION METHOD. - `{ X, Y, Z }` - `P = { C, C++, JAVA }` - `E = { 2, 4, 6, 8, 10, 12, 14, ... }` - `{ 1, A, TED, BETH, PHILIPPINES }` - `{ 1, 2, 3, ... 150 }` ### RULE METHOD - A DESCRIPTIVE PHRASE IS USED TO DESCRIBE THE ELEMENTS OR MEMBERS OF THE SET. IT IS ALSO CALLED SET BUILDER NOTATION, IT IS WRITTEN AS `{X|P(X)}`. - `0 = { X | X IS AN ODD POSITIVE INTEGER LESS THAN 15}` - `R = { X | X IS A REAL NUMBER }` - `V = { X | X IS A VOWEL LETTER }` ## Venn Diagram - DEVELOPED BY JOHN VENN. - METHOD OF USING DIAGRAMS TO ILLUSTRATE THE SET THEORY. - THE UNIVERSAL SET U CONTAINS ALL OBJECTS UNDER CONSIDERATION ## OPERATIONS ON SET ### UNION - The union of two sets A and B, denoted by $A∪B$, is the set that contains all the elements of A and B. ### INTERSECTION - The intersection of two sets A and B, denoted by $A∩B$, is the set that contains all the elements that are common to both A and B. ### COMPLEMENT - The complement of a set A, denoted by $A'$, is the set that contains all the elements in the universal set U that are not in A. ### DIFFERENCE - The difference of two sets A and B, denoted by $A - B$, is the set that contains all the elements that are in A but not in B. ### SYMMETRIC DIFFERENCE - The symmetric difference of two sets A and B, denoted by $A△B$ or $A⊕B$, is the set that contains all the elements that are in A or B, but not in both.
