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Rock-Paper-Scissors-Lizard-Spock Game: Representing Relations, Exercises of Discrete Mathematics

Solutions to problem set #17 of cse 1400 and mth 2051 university courses, focusing on representing relations through graphs and adjacency matrices. The set includes rules for the rock-paper-scissors-lizard-spock game and instructions to construct adjacency matrices for the divides relation and congruence mod 5 relation.

What you will learn

  • Is the Rock-Paper-Scissors-Lizard-Spock game a partial order or an equivalence relation?
  • What is the adjacency matrix of Rock-Paper-Scissors-Lizard-Spock game?
  • How to represent the rules of Rock-Paper-Scissors-Lizard-Spock game using a graph?

Typology: Exercises

2021/2022

Uploaded on 09/12/2022

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In Class Problem Set #17
CSE 1400 and MTH 2051
Fall 2012
Representing Relations
1. Consider the rules of “Rock-Paper-Scissors-Lizard-Spock”
Scissors cuts paper
Paper covers rock
Rock crushes lizard
Lizard poisons Spock
Spock smashes scissors
Scissors decapitates lizard
Lizard eats paper
Paper disproves Spock
Spock vaporizes rock
Rock crushes scissors
(a) Draw a graph which shows the rules of “Rock-Paper-Scissors-
Lizard-Spock”
Answer: Let R, P, Sc, L, Sp stand for Rock, Paper, Scissors,
Lizard, Spock. The graph of rules is
R
P
Sc L
Sp
(b) Construct an adjacency matrix which shows the rules of
“Rock-Paper-Scissors-Lizard-Spock”
Answer: Let R, P, Sc, L, Sp stand for Rock, Paper, Scissors,
Lizard, Spock. The adjacency matrix is
pf3

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CSE 1400 and MTH 2051

Fall 2012

Representing Relations

  1. Consider the rules of “Rock-Paper-Scissors-Lizard-Spock”
  • Scissors cuts paper
  • Paper covers rock
  • Rock crushes lizard
  • Lizard poisons Spock
  • Spock smashes scissors
    • Scissors decapitates lizard
    • Lizard eats paper
    • Paper disproves Spock
    • Spock vaporizes rock
    • Rock crushes scissors

(a) Draw a graph which shows the rules of “Rock-Paper-Scissors- Lizard-Spock” Answer: Let R, P, Sc, L, Sp stand for Rock, Paper, Scissors, Lizard, Spock. The graph of rules is R

P

Sc L

Sp

(b) Construct an adjacency matrix which shows the rules of “Rock-Paper-Scissors-Lizard-Spock” Answer: Let R, P, Sc, L, Sp stand for Rock, Paper, Scissors, Lizard, Spock. The adjacency matrix is

R P Sc L Sp R 1 0 1 1 0 P 1 1 0 0 1 Sc 0 1 1 1 0 L 0 1 0 1 1 Sp 1 0 1 0 1 (c) Is the game a partial order? Why or why not? Answer: The answer is no, the game is not a partial order. I’ve made the game reflexive by my choice. By looking at the adja- cency matrix, you can determine that the game is antisymmet- ric. However, the game is not transitive, which you can see by looking at the graph. For example, Rock smashes Lizara and Lizard poisons Spock, but Rock does not beat Spock: The edges are directed. (d) Is the game an equivalence? Why or why not? Answer: The answer is no, the game is not an equivalence rela- tion. By looking at the adjacency matrix, you can determine that the game is not symmetric and the game is not transitive.

  1. Construct an adjacency matrix for the divides relation on the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Answer:

0 1 2 3 4 5 6 7 8 9 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 2 1 0 1 0 1 0 1 0 1 0 3 0 0 0 1 0 0 1 0 0 1 4 0 0 0 0 1 0 0 0 1 0 5 0 0 0 0 0 1 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0 7 0 0 0 0 0 0 0 1 0 0 8 0 0 0 0 0 0 0 0 1 0 9 0 0 0 0 0 0 0 0 0 1

  1. Construct an adjacency matrix for the congruence mod 5 relation on the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Answer: