School

York UniversityDepartment

Electrical Engineering and Computer ScienceCourse Code

EECS 1019Professor

Suprakash DattaStudy Guide

MidtermThis

**preview**shows half of the first page. to view the full**3 pages of the document.**MATH/EECS 1028 Winter 2017: First test (version 1) – Solutions

Instructor: S. Datta

1. (3 points) Determine, by constructing a truth table, whether the following is a tautology.

(¬p∧(p→q)) → ¬q

Solution: The truth table is given below.

p q p →q¬p∧(p→q) (¬p∧(p→q)) → ¬q

T T T F T

T F F F T

F T T T F

F F T T T

Not a tautology.

2. (3 points) Show that ¬p↔qand p↔ ¬qare logically equivalent.

Solution: The truth table is given below. Since the third and fourth columns are iden-

tical, the two given formulas are logically equivalent.

p q ¬p↔q p ↔ ¬q

T T F F

T F T T

F T T T

F F F F

3. (2 points) Let pbe the proposition “Grizzly bears have been seen in the area”, qbe the

proposition “Hiking is safe on the trail” and rbe the proposition “Berries are ripe on the

trail”. Express the following using propositions p, q, r and logical connectives (including

negation).

(a) (1 point) Grizzly bears have not been seen in the area and hiking on the trail is safe,

but berries are ripe along the trail.

Solution: The statement is ¬p∧q∧r.

(b) (1 point) If berries are ripe along the trail, hiking is safe if and only if grizzly bears

have not been seen in the area.

Solution: The statement is r→(q↔ ¬p).

4. (3 points) Use Venn diagrams to solve the following problem.

An examination of 19 hospital patients turns up 9 cases of fever, 4 cases of jaundice, and

11 cases of fatigue. 7 people have both fever and fatigue, 2 have fever and jaundice, and

2 have jaundice and fatigue. 6 people have none of the three problems. How many people

have just fever or just jaundice?

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