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Unformatted text preview: 2112.2 Set Operations10ifa=candb=d. [Hint:First show that{{a},{a,b}} ={{c},{c,d}}if and only ifa=candb=d.]38.This exercise presentsRussells paradox. LetSbe the setthat contains a setxif the setxdoes not belong to itself,so thatS= {xx/x}.a)Show the assumption thatSis a member ofSleads toa contradiction.b)Show the assumption thatSis not a member ofSleadsto a contradiction.By parts (a) and (b) it follows that the setScannot be defined as it was. This paradox can be avoided by restrictingthe types of elements that sets can have.39.Describe a procedure for listing all the subsets of a finiteset.2.2 Set OperationsIntroductionTwo sets can be combined in many different ways. For instance, starting with the set of mathematics majors at your school and the set of computer science majors at your school, we can formthe set of students who are mathematics majors or computer science majors, the set of studentswho are joint majors in mathematics and computer science, the set of all students not majoringin mathematics, and so on.DEFINITION 1LetAandBbe sets. Theunionof the setsAandB, denoted byAB, is the set that containsthose elements that are either inAor inB, or in both.An elementxbelongs to the union of the setsAandBif and only ifxbelongs toAorxbelongstoB. This tells us thatAB= {xxAxB}.The Venn diagram shown in Figure 1 represents the union of two setsAandB. The areathat representsABis the shaded area within either the circle representingAor the circlerepresentingB.We will give some examples of the union of sets.EXAMPLE 1The union of the sets{1,3,5}and{1,2,3}is the set{1,2,3,5}; that is,{1,3,5} {1,2,3} ={1,2,3,5}.EXAMPLE 2The union of the set of all computer science majors at your school and the set of all mathematicsmajors at your school is the set of students at your school who are majoring either in mathematicsor in computer science (or in both).DEFINITION 2LetAandBbe sets. Theintersectionof the setsAandB, denoted byAB, is the setcontaining those elements in bothAandB.An elementxbelongs to the intersection of the setsAandBif and only ifxbelongs toAandxbelongs toB. This tells us thatAB= {xxAxB}.112 / Basic Structures: Sets, Functions, Sequences, and Sums212UBAA B is shaded.FIGURE 1Venn Diagram Representingthe Union ofAandB.UBAA B is shaded.FIGURE 2Venn Diagram Representingthe Intersection ofAandB.The Venn diagram shown in Figure 2 represents the intersection of two setsAandB. The shadedarea that is within both the circles representing the setsAandBis the area that represents theintersection ofAandB.We give some examples of the intersection of sets.EXAMPLE 3The intersection of the sets{1,3,5}and{1,2,3}is the set{1,3}; that is,{1,3,5} {1,2,3} ={1,3}....
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This note was uploaded on 12/16/2009 for the course MATH 1014 taught by Professor Ganong during the Spring '09 term at York University.
 Spring '09
 ganong

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