# MATH 2112 Discrete Mathematics for Computer Science Questions

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CSCI/MATH 2112
Discrete Structures I
Assignment 1
Due: May 18, 11:59PM Atlantic Standard Time
Note that the set of natural numbers N includes 0.
Problem 1 (4pts): The following sets are written in the set comprehension notation. Please list their
elements explicitly, e.g., {x | x ∈ N, x < 4} = {0, 1, 2, 3}. • {x | x ∈ N, x is even and x is prime} • {x | x ∈ N, x is even and x is odd} • {S | S ∈ Pow({0, 1, 2, 3}), #S = 2} • {A | A ∈ Pow(N), #A = 1} Problem 2 (3pts): The set {2, 4, . . .} is ambiguous. It can mean at least three different sets. Please describe three different sets using the set comprehension notation. Problem 3 (3pts): Find the power sets of the following sets. • {6, 7, 8} • {∅} • {∅, {∅}, {∅, {∅}}} Problem 4 (8pts): Suppose A, B, C are sets. Are the following statements true? If it is true, give an example. If it is not true, give a counter example. • A ∪ (B ∪ C) = (A ∪ B) ∪ C. • A × (B × C) = (A × B) × C. • If A ⊆ B and B ⊆ C, then A ⊆ C. • A − B ⊆ A. 1 Problem 5 (6pts): Suppose U is a set. Let X ⊆ U , we write X c to mean the set {x | x ∈ U, x ∈ / X}. We call X c the complement set of X with respect to U . Now let U = {x | x ∈ N, 0 ≤ x ≤ 10}. Moreover, let S = {4, 3, 6, 7, 1, 9} and T = {5, 6, 8, 4} be subsets of U . Write down the following sets explicitly. Note: the set of natural numbers N includes 0. • Sc • S ∩ Sc • S ∪ Sc • S − Sc • S − Tc • (S ∩ T c )c Problem 6 (1pt): Draw the diagram that represents the relation {(a, a), (a, b), (b, c), (c, b), (c, d), (d, a), (d, b)}. Problem 7 (5pts): Here is a diagram for a relation R on a set A. • Write down the sets A and R explicitly. • Is the relation R reflexive? Symmetric? Transitive? If one of these properties fails to hold, explain why. x y z u v w Problem 8 (4pts) A relation R on a set A such that for all a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a = b is called antisymmetric. Give an example of a relation on a set that is • Both symmetric and antisymmetric. • Neither symmetric nor antisymmetric. You must represent each relation in set notation as well as a directed graph. Problem 9 (4pts): Let A = {a, b, c, d, e}. Find the transitive closure of the following relations on A. You must represent each closure in set notation as well as a directed graph. • {(a, c), (b, d), (c, a), (d, b), (e, d)}. • {(a, b), (a, c), (a, e), (b, a), (b, c), (c, a), (c, b), (d, a), (e, d)}. 2 Problem 10 (2pts): Let A be the set {1, 2, 3}. Find an example of a relation R on A such that it satisfies the following: • Let R1 be the transitive closure of R. • Let R2 be the reflexive closure of R1 . • Let R3 be the symmetric closure of R2 . • R3 is not transitive. You must represent the relation R in set notation as well as a directed graph. 3 Purchase answer to see full attachment Tags: math algebra notation User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

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