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Spring2006test1 - MATH 3012 Sample Quiz Questions for Test...

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MATH 3012 Sample Quiz Questions for Test 1, Spring 2006 WTT Note 1: There are approximately two to three times as many problems listed here as you can expect on an hour exam, but this more comprehensive version should be of greater assistance to students in studying for the test. Note 2 : Often professors give tests with instructions like: Be sure to explain your answers. 1. How many 14-letter words can be formed using the 26 letters of the alphabet if: a. Repetition of letters is allowed. Answer: 26 14 . b. Repetition of letters is not allowed. Answer: P (26 , 14). c. Each word contains exactly 4 vowels, with repetition of letters allowed. Answer: ( 14 4 ) 5 4 21 10 . d. Each word contains exactly 4 vowels, with repetition of letters not allowed. Answer: ( 14 4 ) P (5 , 4) P (21 , 10). 2. Let A = { 1 , 2 , 3 , 4 , 5 } and B = { a, b, c, d, e } . Determine if the following relations are functions. If they are functions, determine if they are surjections, injections or bijections. a. R 1 = { (1 , a ) , (2 , b ) , (3 , d ) , (4 , c ) , (5 , e ) } . Answer: R 1 is a function. It is a bijection. b. R 2 = { (1 , a ) , (2 , b ) , (3 , c ) } . Answer: R 2 is not a function since there is no element y B for which (4 , y ) R 2 , i.e., there is no image of the element 4 A . c. R 3 = { (1 , a ) , (2 , b ) , (2 , c ) , (3 , d ) , (4 , e ) } . Answer: R 3 is not a function since it contains the pairs (2 , b ) and (2 , c ). To be a function from A to B , it is necessary that for each a A , there is only one y B for which ( a, y ) belongs to the relation. 3. Let X = { 1 , 2 , 3 , 4 , 5 } . Which of the following relations are equivalence relations on X ? Which are partial orders on X ? a. R 1 = { (1 , 1) , (2 , 2) , (3 , 3) , (4 , 4) , (5 , 5) } . Answer: R 1 is both a partial order and an equiva- lence relation. Note, as a partial order, R 1 determines an antichain. b. R 2 = { (1 , 1) , (2 , 2) , (3 , 3) , (4 , 4) , (5 , 5) , (2 , 4) , (4 , 3) , (2 , 5) } . Answer: R 2 is neither a partial or- der nor an equivalence relation. It violates the transitive requirement. Note that (2 , 4) , (4 , 3) R 2 but (2 , 3) R 2 . c. R 3 = { (1 , 1) , (2 , 2) , (3 , 3) , (4 , 4) , (5 , 5) , (2 , 4) , (4 , 3) , (2 , 3) , (2 , 5) } Answer: This relation is a partial order but not an equivalence relation. d. R 4 = { (1 , 1) , (2 , 2) , (3 , 3) , (4 , 4) , (5 , 5) , (2 , 4) , (4 , 2) , (5 , 3) , (3 , 5) } . Answer: This relation is an equivalence relation but not a partial order. e. R 5 = { (1 , 1) , (2 , 2) , (3 , 3) , (4 , 4) , (5 , 5) , (2 , 4) , (4 , 2) , (5 , 3) , (3 , 5) , (2 , 3) } Answer: This relation is neither. 4. KK Bakery sells 8 varieties of donuts. Coffee can be ordered black, with milk, with sugar or with both.
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a. How may ways can a customer order a coffee and a donut? Answer: ( 8 1 ) 2 2 . a. How may ways can a customer order a coffee and two donuts? Answer: ( ( 8 1 ) + ( 8 2 ) )2 2 . Note that we allow for the donuts to be the same type. b. If a box of donuts contains a dozen donuts, how many different boxes could be purchased?
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