Question 1(10 points)

Assume A is the set of positive integers less than 4 and B is the set of positive integers less than 5 and R is a relation from A to B and R = {(1, 2), (2, 3), (3, 4)} Which of the following describes this relation?

Question 1 options:

{(a, b) | a A, B B, b = a + 1}

{(a, b) | a A, B B, a > b b > a}

{(a, b) | a A, B B, a b}

{(a, b) | a A, B B, a = b + 1}

Question 2(10 points)

Which of the following ordered pairs doesnotbelong to the relation illustrated by the above arrow diagram?

Question 2 options:

(1, x)

(2, z)

(3, y)

(4, z)

Question 3(10 points)

Assume S = {0, 1, 2, 3}. Which of the following describes the relation defined on S shown in the above directed graph?

Question 3 options:

{(a, b) | a, b S, b = (a + 1) mod 4}

{(a, b) | a, b S, a is even b is odd}

{(a, b) | a, b S, a b}

{(a, b) | a, b S, b = a + 1}

Question 4(10 points)

Given the relation R = {(n, m) | n, m , n m}, which of the following relations defines the inverse of R?

Question 4 options:

R = {(n, m) | n, m , n m}

R = {(n, m) | n, m , n m}

R = {(n, m) | n, m , n = m}

R = {(n, m) | n, m , n > m}

Question 5(10 points)

Given the relation R = {(n, m) | n, m , n < m}. Among reflexive, symmetric, antisymmetric and transitive, which of those properties are true of this relation?

Question 5 options:

It is both reflexive and transitive

It is reflexive, antisymmetric and transitive

It is both antisymmetric and transitive

It is only transitive

Question 6(10 points)

Among reflexive, symmetric, antisymmetric and transitive, which of those properties are true of the above relation?

Question 6 options:

It is both symmetric and transitive

It is only antisymmetric

It is both reflexive and symmetric

It is only symmetric

Question 7(10 points)

Assume S = {a, b, c, d} and R is the relation defined on set S as follows, R = {(a, a), (b, b), (c, c), (a, b), (b, a), (b, c)}. Which of the following ordered pairs must be added to R to form its transitive closure?

Question 7 options:

Only (c, b)

Only (a, c)

Only (d, d)

None, it is already transitive

Question 8(10 points)

Given the relation R = {(n, m) | n, m , n < m}. Which of the following statements about R is correct?

Question 8 options:

R is a partial order

R is not a partial order because it is not transitive

R is not a partial order because it is not reflexive

R is not a partial order because it is not reflexive and not antisymmetric

Question 9(10 points)

Given the relation R on the set S = {0, 1, 2} where R = {(n, m) | n, m S, n + m = 2}. Which of the following statements about R is correct?

Question 9 options:

R is not an equivalence relation because it is neither reflexive nor symmetric

R is not an equivalence relation because it is neither reflexive nor transitive

R is an equivalence relation

R is not an equivalence relation because it is not symmetric

Question 10(10 points)

Assume S = {1, 2, 3}. Consider the equivalence relation R defined on S S, as follows, R = {((x, y), (x, y)) | (x, y), (x, y) S S, x y = x y). How many equivalence classes does that relation have?

Question 10 options:

1

9

6

3

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