Set theory

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By: dvl7 (18 month(s) ago)

heyyy....nice work..!! :)

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SET Theory & Operations An Interactive Powerpoint Presented BY Abhishek Chakraborty

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Set – A collection of objects example : a set of tires Element – An object contained within a set example : M y car’s left front tire

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Finite set – Contains a countable number of objects Example : The car has 4 tires Infinite set - Contains an unlimited number of objects Example: The counting numbers {1, 2, 3, …}

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Cardinal Number : Used to count the objects in a set Example: There are 26 letters in the alphabet Ordinal Number : Used to describe the position of an element in a set Example: The letter D is the 4 th letter of the alphabet

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Equal sets – Sets that contain exactly the same elements (in any order) {A, R, T, S} = {S, T, A, R} Notation: A = B means set A equals set B

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Equivalent sets – Sets that contain the same number of elements (elements do not have to be the same) {C, A, T} ~ {d, o, g} Notation: A ~ B means set A is equivalent to set B

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Empty Set – A set that contains no elements Notation: { } or  Universal Set – A set that contains all of the elements being considered Notation: U

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Complement of a set – A set that contains all of the elements of the universal set that are not in a given set Notation: means the complement of B

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A = {2, 4, 6, 8} B = {1, 2, 3, 4, …} C = {1, 2, 3, 4, 5} D = { } E = {Al, Ben, Carl, Doug} F = { 5, 4, 3, 2, 1} G = {x | x < 6 and x is a counting number} Set Builder Notation Which sets are finite? Which sets are equal to set C? Which sets are equivalent to set A? {1, 2, 3, 4, 5} n(E) = n(G) = 4 5 F, G E A, C, D, E, F, G

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Is { } the same as ?  Yes Is { } the same as ?   No 

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Set B is a subset of set A if every element of set B is also an element of set A. Notation: B  A W = {1, 2, 3, 4, 5} X = {1, 3, 5} Y = {2, 4, 6} Z = {4, 2, 1, 5, 3} True or False: X  W Y  W Z  W   W True False True True * The empty set is a subset of every set

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Set B is a proper subset of set A if every element of set B is also an element of set A AND B is not equal to A. Notation: B  A W = {1, 2, 3, 4, 5} X = {1, 3, 5} Y = {2, 4, 6} Z = {4, 2, 1, 5, 3} True or False: X  W Y  W Z  W True False False

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How many subsets can a set have? Set {a} {a, b} {a, b, c} Number of Elements 1 2 3 n Subsets Number of Subsets 2 4 8 2 n {a} ,{ } {a} ,{b} ,{ a,b } ,{ } {a} ,{b} ,{c} ,{ a,b }, { a,c } ,{ b,c } ,{ a,b,c }, { } If a set has n elements, it has 2 n subsets

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How many proper subsets can a set have? Set {a} {a, b} {a, b, c} {a, b, c, d} Number of Elements 1 2 3 4 n Proper Subsets Number of Proper Subsets 1 3 7 15 2 n – 1 {a} ,{ } {a} ,{b} ,{ a,b } ,{ } {a} ,{b} ,{c} ,{ a,b }, { a,c } ,{ b,c } ,{ a,b,c }, { } If a set has n elements, it has 2 n – 1 proper subsets X X X

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A Venn Diagram allows us to organize the elements of a set according to their attributes. Wings Horn HORSE – bred for magical Process PEGASYS UNICORN

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U = {1, 2, 3, 4, 5, 6.5} even odd prime 1 2 3 4 5 6.5 Venn Diagram

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National Library of Virtual Manipulatives Attribute Blocks small blue triangle

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Set Operations The intersection of sets A and B is the set of all elements in both sets A and B notation: A  B

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The union of sets A and B is the set of all elements in either one or both of sets A and B notation: A  B

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The union of sets A and B is the set of all elements in either one or both of sets A and B notation: A  B

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A = {1, 2, 3, 4, 5} B = {2, 4, 6} C = {3, 5, 7} A  B = A  B = C  B = C  B = {2, 4} {1, 2, 3, 4, 5, 6} {2, 3, 4, 5, 6, 7) { } The set complement X – Y is the set of all elements of X that are not in Y A – B = C – A = {1, 3, 5} {7}

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Representing sets with Venn diagrams A B A B C Three attributes 2 3 or 8 regions 1 2 3 4 1 2 3 4 5 6 7 8 Two attributes 2 2 or 4 regions

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A B A A B A

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A B A U B A B A  B A B A  B

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A B A U B A B A  B A B C (A U B)  C A B C (A U B)  C

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(A U B)  C A B C 1 2 3 4 5 6 7 8 A = B = C = C = A U B = (A U B)  C = {1, 2, 4, 5} {2, 3, 5, 6} {4, 5, 6, 7} {1, 2, 3, 8} {1, 2, 3, 4, 5, 6} {1, 2, 3}

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A U (B  C) A B C A = B = C = B  C = A U (B  C) = {3, 6, 7, 8} {2, 3, 5, 6} {4, 5, 6, 7} {5, 6} {3, 5, 6, 7, 8} 3 7 8 6 5 4 1 2

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A B 1 2 3 4 How many stars are in: Circle A Circle B Only Circle A Both A and B Either A or B Exactly one circle Neither circle Total stars = 3 5 2 1 7 6 2 9

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B F Out of 20 students: 8 play baseball 7 play football 3 play both sports How many play neither sport? How many play only baseball? How many play exactly one sport? 20 3 5 4 8 8 5 5 + 4 = 9

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B P G Out of 30 people surveyed: 20 like Blue 20 like Pink 15 like Green 14 like Blue and Pink 11 like Pink and Green 12 like Blue and Green 10 like all 3 colors How many people like only Pink? How many like Blue and Green but not Pink? How many like none of the 3 colors? How many like exactly two of the colors? 10 2 1 4 2 5 4 2 30 5 2 2 4 + 2 + 1 =7