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# Exam2 - Exam2 Voting Systems p 3(IIA counting principles...

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Exam2: Voting Systems p. 3(IIA, counting principles plus small portion of chapter 11) SETS AND COUNTING PRINCIPLES 1) Describe what a set is and how sets are denoted. 2) Give examples of collections that are sets. 3) Give examples of collections that are not sets. 4) Describe empty sets, the associated symbols, and give examples of empty sets. 5) Describe subsets and give examples. 6) Define the cardinality of a set, its symbolic form, and give examples. The number of elements contained in a set is called the cardinality. If A reperesents a set then the cardinality of A is written n(A) Ex: A = {1,5,7,9} Then n(A) = 4 7) Define power set and its cardinality.

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The collection of all subsets of a set is called power set. If a finite set contains n elements then the power set contains 2n. Ex: A = {1.5.7.9} has 2 4 = 16 subsets Empty and original sets are always subsets. 8) Write the members of the power set of the sets (i.e., list all possible subsets of): a) {A, B} b) {A, B, C} c) {A, B, C, D} Subsets: φ , {A} {B} {A,B} d) {A, B, C, D, E} Do part d) for HW. Results published online. SUBSETS: φ , {A}, {B}, {C}, {D}, {E}, {A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B,D}, {B,E}, {C,D}, {C,E}, {D,E}, {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E), {A,D,E}, {B,C,D}, {B,C,E}, {B,D,E}, {C,D,E}, {A,B,C,D}, {A,B,C,E}, {A,B,D,E}, {A,C,D,E}, {B,C,D,E}, {A,B,C,D,E}. There are subsets. 9) Fundamental multiplication principle of mathematics a) AN EXAMPLE: Consider the sets E = {a, b, c} and F = {1, 2}. Write out all possible pairs of elements in such a way that the first element is from set E and the
second element is from set F. Construct a tree diagram and a matrix. Pairs: (a,1) (a,2) (b,1) (b,2) (c,1) (c,2) TREE DIAGRAM ORDERED PAIRS MATRIX E 1 2 a (a,1) (a,2) b (b,1) (b,2) c (c,1) (c,2) b) Statement: For two events , that occur in succession. If event E1 occurs in m different ways and (independently of E1) a sexond events E2 occurs in n different ways, then the two events occur in a total of m.n 10) A person wants to purchase a cellular phone and a calling plan. Suppose that there are two choices of cellular phones (the Motorola and the Nokia) and three choices of calling plans (one for \$29.99 which allows 300 minutes of

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airtime per month, a second for \$39.99 which allows 600 minutes of airtime per month, and a third for \$49.99 which allows 1000 minutes of airtime per month). In how many different ways can this person purchase a cellular phone and a calling plan? Show all possibilities using a matrix and using a tree diagram. Cell Phones 29.99 39.99 49.99 M (M, 29.99) (M,39.99) (M,49.99) N (N, 29.99) (N, 39.99) (N,49.99) # of possibilites = (2) (3) = 6 11) A game consists of tossing a coin and then rolling a die. How many different outcomes are possible? Show all possible outcomes using a matrix and using a tree diagram. Possibilities = (2) (6) = 12
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Exam2 - Exam2 Voting Systems p 3(IIA counting principles...

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