Applied Discrete Math Quiz 3

Consider the following partial grade book for a certain class. Name | Exam 1 Score ------------------------- Bill | 70 Gene | 83 Jesse | 86 ------------------------- Consider the set A = {Bill, Gene, Jesse} and B = {0,1,... , 100}. Then consider the relation R from A to B consisting of pairs (a,b) from the table above with a ∈ A, b ∈ B. In listing all the pairs of R, we have {(70,Bill),(83,Gene),(86,Jesse)}. True False

False

Let A = {0,1,2} and B = {4,8,9}. Let R be the relation from A to B such that (a,b) ∈ R provided that ab = 8. Then all the pairs in R are {(8,1),(4,2)} True False

False

Let A = {0,2,4} and B = {0,1,2}. Let R be the relation from A to B of all ordered pairs of the form (a,b), where a is in A and b is in B, and where a/b is an integer. Then the pairs in R are {(0,1),(2,1)(2,2),(4,1)}. True False

False

Let A = {1,2,3}, B = {2,3}, and C = {3,6}. Suppose R is a 3-ary relation that is defined by the following rule (a,b,c) is an element of R provided that a + 2 = b and b | c, and where a ∈ A, b ∈ B, c ∈ C. Then the triples in R are {(1,3,6)}. True False

False

Let A = {1,2,7} and consider the relation R on the set A defined by R = {(1,2), (1,7), (7,1)}. In representing R using a digraph, there should be a directed edge from 2 to 1. True False

False

Let A = {0,1,2} and R be the relation on A given by R = {(0,0),(1,1), (2,2)}. Then R is symmetric. True False

True

Let A = {0,1,2} and R be the relation on A given by R = {(0,0),(1,1),(2,2)}. Then R is transitive. True False

True

Let A = {3,4,5}, and let R be the relation on A consisting of pairs of the form (a,b), where ab > 15, and both a and b are elements of A. If we represent R using a matrix (where the first, second, and third rows/columns of the matrix correspond to 3, 4, and 5, respectively), the matrix would be: [ 0 0 0 ] 0 1 1 [ 0 1 1 ] True False

True

Let A = {0,1,2} and R be the relation on A given by R = {(0,0),(0,1),(0,2),(1,2),(2,2)}. Is R antisymmetric? No Yes

Yes

Let A = {0,1,2} and R be the relation on A given by R = {(0,0),(0,1),(1,0)}. Is R symmetric? Yes No

Yes

Let A = {0,2,4} and let R be the relation from A to A consisting of pairs (a,b), where a ≤ b. List all the pairs in R. {(0,0),(2,0),(2,2),(4,0),(4,2),(4,4)} {(0,0),(2,2),(4,4)} {(0,0),(0,2),(0,4),(2,2),(2,4),(4,4)}. {(2,0),(4,0),(4,2)}

{(0,0),(0,2),(0,4),(2,2),(2,4),(4,4)}.

Let A = {1,2,3} and B = {0,1,2,3}. If R is the "<" relation from A to B, list the pairs in R. {(0,1),(0,2),(1,2),(0,3),(1,3)(2,3)} {(2,1)(3,1)} {(1,2),(1,3),(2,3)} {(1,0),(2,0),(2,1),(3,0),(3,1),(3,2)}

{(1,2),(1,3),(2,3)}

Let A = {3,6,9}, and let R be the relation on A such that (a,b) ∈ R provided a | b, where a and b are both elements of A. List all pairs in R. No pairs in this relation {(3,3),(6,6),(6,3),(9,9),(9,6),(9,3)} {(3,3),(6,6),(9,9)} {(3,3),(3,6),(3,9),(6,6),(9,9)}

{(3,3),(3,6),(3,9),(6,6),(9,9)}

Let A = {0,1,2} and R be the relation on A given by R = {(0,0),(0,1),(0,2),(1,0),(1,2),(2,1),(2,2)}. Is R reflexive? Yes No

No

Let A = {0,1,2} and R be the relation on A given by R = {(0,0),(0,1),(1,0),(1,2),(2,1),(2,2)}. Is R transitive? No Yes

No