Set Theory Problems Suppose two dice are rolled. Answer the following questions

Question

Set Theory Problems

Suppose two dice are rolled. Answer the following questions and illustrate your answers. a) What is the universal set of this experiment? b) Set A is the set in which the number on the first die is not less than the number on the second die. What is set A? c) What is set B corresponding to the number on the first die be 5? d) What is set C corresponding to the sum of the numbers on the two dice be 7? e) What is set D corresponding to the sum of the numbers on the two dice be less than 5? f) What is set E corresponding to the numbers on the two dice differ by 2? Two cars are approaching a crossroad from different directions. One car is driven from north to south, the other from east to west. Suppose the first car arrives at the crossroad at time t_1 and the second car at time t_2, where 0 lessthanorequalto t_1 lessthanorequalto T and 0 lessthanorequalto t_2 lessthanorequalto T. a) What is the universal set of arrivals of the two cars at the crossroad? b) Determine and illustrate the set correspond to car one arrives at the crossroad between times 1 and 2, and car 2 arrives between times 2 and 3. c) Suppose each car needs tau seconds to pass the crossroad. Determine and illustrate the set corresponding to collision between the two cars. d) Determine and illustrate the set corresponding to a collision being missed by 1 second or less.

Explanation / Answer

Solution:-

Two dice are rolled.

a) Universal set, S =

b) Set A, where number on first die is not less than the second die.

A = {(1,1)(2,1)(2,2)(3,1)(3,2)(3,3)(4,1)(4,2)(4,3)(4,4)(5,1)(5,2)(5,3)(5,4)(5,5)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)}

c) Set B, number on first die be 5.

B = {(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)}

d) Set C, sum of the two dice be 7.

C = {(1,6)(2,5)(3,4)(4,3)(5,2)(6,1)}.

e) Set D, sum on two dice being less than 5.

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

f) Set E, number differ by 2.

E = {(1,3)(2,4)(3,1)(4,6)(6,4)}

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

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