COMP2111 1 2020 Term 1

Java Python COMP2111

Assignment 1

2020 Term 1

Due: Thu, 19th March, 12:00 noon

Problem 1 (20 marks)

Recall the relation composition operator ; defined as:

R1; R2 = {(a, c) | there is a b with (a, b) ∈ R1 and (b, c) ∈ R2}

For any set S, and any binary relations R1, R2, R3 ⊆ S × S, prove or give a counterexample to disprove the following:

(a) (R1; R2); R3 = R1;(R2; R3)                                                       (4 marks)

(b) I; R1 = R1; I = R1 where I = {(x, x) | x ∈ S}                             (4 marks)

(c) (R1; R2)← = R1 ←; R2 ←                                                         (4 marks)

(d) (R1 ∪ R2); R3 = (R1; R3) ∪ (R2; R3)                                        (4 marks)

(e) R1;(R2 ∩ R3) = (R1; R2) ∩ (R1; R3)                                           (4 marks)

Problem 2 (30 marks)

Let R ⊆ S × S be any binary relation on a set S. Consider the sequence of relations R 0 , R 1 , R 2 , . . ., defined as follows:

R0 := I = {(x, x) | x ∈ S}, and

Ri+1 := Ri ∪ (R; Ri) for i ≥ 0

(a) Prove that if there is an i such that R i = R i+1 , then R j = R i for all j ≥ i. (4 marks)

(b) Prove that if there is an i such that R i = R i+1 , then R k ⊆ R i for all k ≥ 0. (4 marks)

(c) Let P(n) be the proposition that for all m ∈ N: R n ; R m = R n+m. Prove that P(n) holds for all n ∈ N. (8 marks)

(d) If |S| = k, explain why R k = R k+1 . (Hint: Show that if (a, b) ∈ R k+1 then (a, b) ∈ R i for some i < k + 1.) (4 marks)

(e) If |S| = k, show that R k is transitive. (4 marks)

(f) If |S| = k, show that (R ∪ R←) k is an equivalence relation. (6 marks)

Problem 3 (22 marks)

Let PF denote the set of well-formed propositional formulae made up of propositional variables, > , ⊥, and the connectives ¬, ∧, and ∨.

We define the function dual(ϕ) from PF to PF, which swaps ∧ and ∨, as well as > with ⊥. We also define flip(ϕ) from PF to PF, which negates adai 写COMP2111 Assignment 1 2020 Term 1 ny propositional variables in the formula:

dual(p) = p                                             flip(p) = ¬p

dual(> ) = ⊥                                          flip(> ) = >

dual(⊥) = >                                           flip(⊥) = ⊥

dual(¬ϕ) = ¬ dual(ϕ)                               flip(¬ϕ) = ¬flip(ϕ)

dual(ϕ ∧ ψ) = dual(ϕ) ∨ dual(ψ)               flip(ϕ ∧ ψ) = flip(ϕ) ∧ flip(ψ)

dual(ϕ ∨ ψ) = dual(ϕ) ∧ dual(ψ)               flip(ϕ ∨ ψ) = flip(ϕ) ∨ flip(ψ)

(a) For the formula ϕ = p ∨ (q ∧ ¬r):

(i) What is dual(ϕ)? (4 marks)

(ii) What is flip(ϕ)? (4 marks)

(b) Prove that for all ϕ ∈ PF: flip(ϕ) is logically equivalent to ¬dual(ϕ). (14 marks)

Problem 4 (28 marks)

Four wifi networks, Alpha, Bravo, Charlie and Delta, all exist within close proximity to one another as shown below.

Networks connected with an edge in the diagram above can interfere with each other. To avoid interference networks can operate on one of two channels, hi and lo. Networks operating on different channels will not interfere; and neither will networks that are not connected with an edge.

Our goal is to determine (algorithmically) whether there is an assignment of channels to networks so that there is no interference. To do this we will transform. the problem into a problem of determining if a propositional formula can be satisfied.

(a) Carefully defining the propositional variables you are using, write propositional formulae for each of the following requirements:                     (4 marks)

(i) ϕ1: Alpha uses channel hi or channel lo; and so does Bravo, Charlie and Delta. (4 marks)

(ii) ϕ2: Alpha does not use both channel hi and lo; and the same for Bravo, Charlie and Delta.(4 marks)

(iii) ϕ3: Alpha and Bravo do not use the same channel; and the same applies for all other pairs of networks connected with an edge. (4 marks)

(b) (i) Show that ϕ1 ∧ ϕ2 ∧ ϕ3 is satisfiable; so the requirements can all be met. Note that it is sufficient to give a satisfying truth assignment, you do not have to list all possible combinations