Problem 1
I’m sure you have all encountered the concept of a set, such as {1,2,5} or the set of all positive rational numbers. In each of the four parts of Problem 1, in order that the problem be completely defined, it is necessary to specify a universe of discourse, which is a set to which the statement in the problem applies.
The universes of discourse for the four parts are:
(a) The set of all integers
(b) The set of all students enrolled at UMUC
(c) The set of all integers
(d) The set of students in this section of CMSC 150
The remaining problems do not require clarification.
1. Which of the following sentences are logical statements? (20 points, 4 parts- 5 points each)
a. If x even divides y, then x is a factor of y
b. If John does well in discrete math, then he will be an excellent programmer
c. 2 is the only even prime number
d. He is the best student in the class
2. Construct the truth tables for the following propositions: (20 points, 4 parts-5 points each)
a. (p ∧ ¬ p) ∧ q
b. (p ∨ q) ∧ (q ∨ ¬ p)
c. p ∧ (q ∨ ¬ r)
d. (p ∧ q) ∨ (p ∧ r)
3. Refer to the propositions in problem 2. For each of them, indicate whether it is a tautology, a contradiction or neither. (20 points, 4 parts – 5 points each)
4. Use truth tables to determine whether the following is valid argument: (20 points)
p → q
q → p
∴ p ∨ q
5. Use truth tables to determine whether each of the following pairs of propositions are logically equivalent. (20 points, 2 parts - 10 points each)
a. (p ∨ q) ∧ ¬ q
¬ q ∧ (q ∨ p)
b. (¬ (p ∧ q)) ∨ q
(¬ p ∧ ¬ q) ∨ q