Math 1090 Homework 1 Math 1090 Homework 1
Solutions

1. In translating each of the following to symbolic form, the word ``or'' would be replaced by Ú or \veebar . What is the appropriate choice in each case? Explain why your choice is correct.

1. To take discrete mathematics, you must have taken calculus or a course in computer science.
Answer: Ú since it is fine if you have taken both.
2. Dinner for two includes two items from column A or three items from column B.
Answer: \veebar since if you choose two from A you cannot in addition choose three from B.
3. When you buy a new car from Acme Motor Company, you get \$ 2000 back in cash or a 2% car loan.
Answer: \veebar since you get one or the other but not both.
4. School is closed if more than 2 feet of snow falls or if the wind chill is below -50.
Answer: Ú since school is closed if both happen to be the case.

2. Determine whether
 ( ~ q Ù(p ® q)) ®   ~ p
is a tautology.

 p q ( ~ q Ù (p ® q)) ® ~ p T T F F T T F T F T F F T F F T F F T T T F F T T T T T

Yes it is a tautology.

3. Does p ® q logically imply (p Ùq) Ú ~ p ? Justify your answer.

 p q p ® q (pÙq) Ú ~ p T T T T T F T F F F F F F T T F T T F F T F T T

Yes. Whenever p® q is T, so is (pÙq) Ú ~ p.

4. Use truth tables to show that both
 p Ú(p Ùq)  and  p Ù(p Úq)
are equivalent to p.

 p q p Ú (pÙq) p Ù (p Úq) T T T T T T T T T F T T F T T T F T F F F F F T F F F F F F F F

5. Determine whether
 (p Úq) ® r   and  p Ú(q ® r )
are equivalent.
 p q r (pÚq) ® r p Ú (q® r) T T T T T T T T T T T F T F F T T F T F T T T T T T T T F F T F F T T T F T T T T T F T T F T F T F F F F F F F T F T T F T T F F F F T F F T T

Their truth tables are different. They are not equivalent.

1. Determine whether
 (p « q) « r   and  p « (q « r )
are equivalent.
2. Is ((p « q) « r)  «   (p « (q « r )) a tautology? Explain.