Homework 01 Solution

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1.1.3) Assume the propositions p, q, r, and s have the following truth values:

  • p is false

  • q is true

  • r is false

  • s is true

What are the truth values for the following compound propositions?

  1. ¬p

  2. p r

  3. q s

  4. q s

  5. q s

  6. q r

1.1.4) Indicate whether each statement is true or false, assuming that the “or” in the sentence means the inclusive or. Then indicate whether the statement is true or false if the “or” means the exclusive or.

(a) February has 31 days or the number 5 is an integer.

(b) The number π is an integer or the sun revolves around the earth.

(c) 20 nickels are worth one dollar or whales are mammals.

(d) There are eight days in a week or there are seven days in a week.

(e) January has exactly 31 days or April has exactly 30 days.

1.2.2) Express each statement in logic using the variables:

  • p: It is windy.

  • q: It is cold.

  • r: It is raining.

(a) It is windy and cold.

(b) It is windy but not cold.

(c) It is not true that it is windy or cold.

(d) It is raining and it is windy or cold.

(e) It is raining and windy or it is cold.

(f) It is raining and windy but it is not cold.

1.2.4) Write a truth table for each expression.

(a) ¬p q

(b) ¬(p q)

(c) r (p ¬q)

(d) (r p) (¬r ¬q)

1.2.7) Consider the following pieces of identification a person might have in order to apply for a credit card:

  • B: Applicant presents a birth certificate.

  • D: Applicant presents a driver’s license.

  • M: Applicant presents a marriage license.

Write a logical expression for the requirements under the following conditions:

(a) The applicant must present either a birth certificate, a driver’s license or a marriage license.

(b) The applicant must present at least two of the following forms of identification: birth certificate, driver’s license, marriage license.

(c) Applicant must present either a birth certificate or both a driver’s license and a marriage license.

1.3.2) Give the inverse, converse and contrapositive for each of the following statements:

(b) If he trained for the race, then he finished the race.

(c) If the patient took the medicine, then she had side effects.

1.3.5) Define the following propositions:

  • c: I will return to college.

  • j: I will get a job.

Translate the following English sentences into logical expressions using the definitions above:

(a) Not getting a job is a sufficient condition for me to return to college.

(b) If I return to college, then I won’t get a job.

(c) I am not getting a job, but I am still not returning to college.

(d) I will return to college only if I won’t get a job.

(e) There’s no way I am returning to college.

(f) I will get a job and return to college.

1.3.7) Define the following propositions:

  • s: a person is a senior

  • y: a person is at least 17 years of age

  • p: a person is allowed to park in the school parking lot

Express each of the following English sentences with a logical expression:

(a) A person is allowed to park in the school parking lot only if they are a senior and at least seventeen years of age.

(b) A person can park in the school parking lot if they are a senior or at least seventeen years of age.

(c) Being 17 years of age is a necessary condition for being able to park in the school parking lot.

(d) A person can park in the school parking lot if and only if the person is a senior and at least 17 years of age.

(e) Being able to park in the school parking lot implies that the person is either a senior or at least 17 years old.

1.3.10) The variable p is true, q is false, and the truth value for variable r is unknown. Indicate whether the truth value of each logical expression is true, false, or unknown.

(a) p → (q r)

(b) (p r) → r

(c) (p r) ↔ (q r)

(d) (p r) ↔ (q r)

a. (true AND unknown) iff (false OR unknown)

b. unknown iff false

(e) p → (r q)

(f) (p q) → r