Assignment 1
Total Points = 50
1. (2+2+2=6 points) In each of the following cases, find a proposition with the given conditions.
(a) Find a proposition with three variables p,q, and rthat is true when pand rare true
and qis false, and false otherwise.
(b) Find a proposition with three variables p,q, and rthat is true when at most one of the
three variables is true, and false otherwise.
(c) Find a proposition with three variables p,q, and rthat is never true.
2. (2+2=4 points) In each of the following cases, write an equivalent proposition as stated.
(a) Write a proposition equivalent to p∨ ¬qthat uses only p, q, ¬, and the connective ∧.
(b) Write a proposition equivalent to p→qusing only p, q, ¬, and the connective ∨.
3. (6 points) Prove that (q∧(p→ ¬q)) → ¬pis a tautology using propositional equivalence
and the laws of logic.
4. (2*4=8 points) Using cfor “it is cold”, dfor “it is dry”, rfor “it is rainy”, and wfor “it is
windy”, write the following sentences in symbols.
(a) It is neither cold nor dry.
(b) It is rainy if it is not cold.
(c) To be windy it is necessary that it be cold.
(d) It is rainy only if it is windy and cold.
5. (4+4=8 points) There are three kinds of people living on an island: knights who always tell
the truth, knaves who always lie, and spies who can either tell the truth or lie. You encounter
three people, A,B, and C. You know one of the three people is a knight, one is a knave,
and one is a spy. Each of the three people knows the type of person each of the other two is.
How would you identify who is who, if:
(a) Asays “I am not a knight,” Bsays “I am not a spy,” and Csays “I am not a knave.”
(b) Asays “I am a spy,” Bsays “I am a spy” and Csays “Bis a spy.”
6. (6 points) Determine whether the following argument is valid. Justify your answer by showing
which rules of inferences have been followed at each step.
She is a Math Major or a Computer Science Major.
If she does not know discrete math, she is not a Math Major.
If she knows discrete math, she is smart.
She is not a Computer Science Major.
Therefore, she is smart.
7. (6 points) Give a direct proof of the theorem: “if xand yare odd integers, then x+yis
even.”
8. (6 points) Give a proof by contradiction of the following: “If xand yare even integers, then
xy is even”.
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