CMSC 203 Spring 2008 Homework Assignment 4 ...

CMSC 203 Spring 2008 Homework Assignment 4Due May 13, 2008Name______________________Show all work!1. Suppose I have a large collection of red, blue, yellow, brown, gray, green, orange, black, tan, white, purple,and pink marbles.(a) How many ways can I choose 80 marbles?(S)lots = 80, (C)ategories = 12, Total (R)estrictions = 0, so the number of choices = C(S+C−R−1,S−R)= C(80 + 12 − 1, 80) = C(91,80).(b) How many ways can I choose 80 marbles if I must have at least 3 of each color in each selection?(S)lots = 80, (C)ategories = 12, Total (R)estrictions = 0, so the number of choices = C(S+C−R−1,S−R)= C(80 + 12 − (3)(12) − 1,80 − (3)(12)) = C(92 − 37,80 − 36) = C(65,44)

CMSC 203 Spring 2008 Homework Assignment 4Due May 13, 2008Name___________________________Show all work!2. Consider the following sets with the corresponding number of elements indicated in each region:AB10 5 14 8(a) Find P(A ∪ B)P(A ∪ B) = |A ∪ B| / |U| = (10 + 5 + 14) / (10 + 5 + 14 + 8) = 29/37.(b) Find P(B | A)P(B | A) = |B ∩ A| / |A| = 5/(10 + 5) = 5/15 = 1/3.

CMSC 203 Spring 2008 Homework Assignment 4Due May 13, 2008Name___________________________Show all work!3. Determine if the probability that a pair of fair dice roll an 7 is independent of the first die being a 2.2nd Die1 2 3 4 5 61 | 2 3 4 5 6 72 | 3 4 5 6 7 81st Die 3 | 4 5 6 7 8 94 | 5 6 7 8 9 105 | 6 7 8 9 10 116 | 7 8 9 10 11 12P((Roll = 7) | (1st Die = 2)) = |(Roll = 7 and 1st = 2)|/|(1st = 2)| = 1/6.P(Roll = 7) = 6/36 = 1/6.Since P((Roll = 7) | (1st Die = 2)) = 1/6 = P((Roll = 7)), they areindependent events.

CMSC 203 Spring 2008 Homework Assignment 4Due May 13, 2008Name___________________________Show all work!4. Consider the Equivalence Relation R on the set, A = {a, b, c, d, e, f, g, h, i, j} described by the graph:bcdefagjih(a) Find [e][e] = {e, b, h}(b) List the elements of the partition of the set A induced by the relation R.Partition sets = {a, c, j} , {b, e, h} , {d, f, i}, {g}.

CMSC 203 Spring 2008 Homework Assignment 4Due May 13, 2008Name___________________________Show all work!5. Let F be a function on the Integers given by F(n) = |n − 2|.(a) Show that the relation R = {(x,y) | x,y are integers and F(x) = F(y)}is a Reflexive, Symmetric,and Transitive relation.Let x be an Integer. Then |x − 2| = |x − 2|, hence (x,x) is in R, so R is Reflexive.Let x and y be Integers with (x,y) in R. Then |x − 2| = |y − 2|, hence |y − 2| = |x − 2|, implying that(y,x) is in R, so R is Symmetric.Finally, let x, y, and z be Integers with (x,y) and (y,z) in R. Thus, |x − 2| = |y − 2| and |y − 2| = |z − 2|.This implies that |x − 2| = |z − 2|, hence (x,z) is in R This make R a Transitive relation.(b) Describe the partition of the Integers induces by R.0 = |2 − 2|, 1 = |3 − 2| = |1 − 2|, 2 = |4 − 2| = |0 − 2|, 3 = |5 − 2| = |(−1) − 2|, 4 = |6 − 2| = |((−2) −2|, etc.So, the Partition = {2}, {1, 3}, {0, 4}, {−1, 5}, {−2,6}, {−3, 7}...

CMSC 203 Spring 2008 Homework Assignment 4Due May 13, 2008Name___________________________Show all work!6. Consider the database consisting of the following Fields and Records:Color Model Length Width Height Weight AgeRed Standard 42 14 24 50 3Red Standard 42 14 24 50 16Red Deluxe 48 18 30 64 8Blue Standard 42 14 24 50 12Blue Standard 42 14 24 50 9Blue Deluxe 48 18 30 64 14Green Standard 42 14 24 50 5Green Deluxe 48 18 30 64 11Green Deluxe 48 18 30 64 6Black Standard 42 14 24 50 4Black Deluxe 48 18 30 64 7Black Deluxe 48 18 30 64 2(a) For this instance of the database, which Fields would serve as Primary Keys?Age.(b) Find P2,3,5P2,3,5 = {(Standard, 42, 24), (Deluxe, 48, 30)}