UMD MATH 111 - The Number of Elements in a Set (3 pages)

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The Number of Elements in a Set



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The Number of Elements in a Set

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Pages:
3
School:
University of Maryland, College Park
Course:
Math 111 - Introduction to Probability

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Math 111 section 6 2 The Number of Elements in a Set notes by Tim Pilachowski Definition A set is a well defined collection of objects The individual objects in a set are called the elements or members of the set The number of elements in a set S sometimes called the cardinality of the set is designated by n S Examples A 6 1 Examples A revisited Let G the set containing the letters x A and the integers 0 9 12 0 x 9 12 A H the set of colors of lights in a standard traffic signal red yellow green I x x2 4 J the set of positive even numbers 2 4 6 8 n G n H n I n J n n Examples B 6 1 Examples G revisited Consider the universe U a b c d e f g h i j k and sets M a b c d N b c d e f g and P g h i n M n N n P M N Mc n Mc M N n M N M N n M N M P n M P P M P n M P Pc n Pc N Pc n N Pc N Pc n N Pc When considering the number of elements in a union one must do more than add the individual cardinalities In the example above the number of elements in M N is less than the sum of the cardinalities of M and N Note that the elements b c and d are included when counting the number of elements in M then again when counting the number of elements in N But since the elements b c and d are only included once each in the union we must find a way to discount the duplication when counting the cardinality of the union The mathematical relationship for the cardinality of the union of any two sets A and B is given in the formula n A B n A n B n A B This is an important formula Remember it We ll need it for probability In terms of the example above n M N and n N P c The case of disjoint sets is just a particular application of the formula above n M P If we know any three of the cardinalities in the formula n A B n A n B n A B we can find the fourth Examples C a n A 20 n A B 6 n A B 30 n B b n C 18 n D 15 n C D 40 n C D Example D is on the next page Example D Among the 178 members of a freshman class at Matriarch University U Mama 37 have academic scholarships 55 are athletes 62 live on campus 14



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