CS 281 Test I

February 16, 1999

1. Express the assertions given below as a proposition of a predicate logic using the predicates: D(x): x is difficult,

E(x): x is expensive,

I(x): x is interesting. The universe is the set of books.[16]

(a) Some books are expensive but interesting.
(b) Not all books are difficult and expensive.
(c) Rosen's book (denote it by Rosen) is not difficult.
(d) No book is expensive.

2. Fill in the blanks with the shortest string of characters so that the resultant proposition is valid. [25]

(a) $\neg (\neg P \wedge Q) \Leftrightarrow (\neg Q$ $\framebox [1.0in]{?}$ P)
(b) $(\neg P \vee (Q \rightarrow P) ) \Leftrightarrow (\neg P \vee ($ $\framebox [1.0in]{?}$ $\vee P) ) \Leftrightarrow$ $\framebox [1.0in]{?}$
(c) $( (P \vee Q) \wedge$ $\framebox [1.0in]{?}$ $) \rightarrow P$
(d) $\neg (P \rightarrow Q) \Leftrightarrow (P$ $\framebox [1.0in]{?}$ $\neg Q)$

3. State the following sentences as an if-then statement, and find their contrapositive, all in English: [24]

(a) Healthy body follows from healthy diet.
(b) Free discussions are necessary for further technological advances.
(c) Fuel can be saved only if there is good insulation or storm window throughout.

4(a) Express the argument given below using a symbol sggested for each proposition. [10]
(b) Using the symbols of (a) for the propositions, explain how the reasoning proceeds i.e. identify each application of inference rule in the argument. Is the reasoning correct ? Give your reasons. [15]

If there is a lot of sun (S), then the crop is good (C). Either there is a lot of sun or there is a lot of rain (R). However, the crop is not good. Therefore there is a lot of rain.

(c) Repeat (a) and (b) for the following argument: [10]

If today is Tuesday (T), then I have a test in Computer Science (C) or a test in Biology (B). But if there is a test in Biology, then it is not Tuesday. Also if my Computer Science teacher is sick (S), then I will not have a test in Computer Science. Today is Tuesday. Therefore my Computer Science teache is sick.