CS 281 Test I

February 17, 1998

1. Express the assertions given below as a proposition of a predicate logic using the following predicates. The universe is the set of numbers.[25]

I(x): x is an integer.
E(x): x is even.
N(x): x is a natural number.
G(x,y ) : x > y

(a) 7 is not an integer.
(b) There are even numbers among the natural numbers.
(c) Not all numbers are a natural number.
(d) A number is even only if it is integer.
(e) For an integer to be a natural number, it is necessary for it to be nonnegative (i.e. not less than 0).

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

(a)
(b)
(c)
(d) .

3. Find the converse and the contrapositive of the following propositions: [24]
(a) Seeing is sufficient for believing.
(b) I will buy it only if it is less expensive.
(c) I can't finish the job, if I don't work harder.
(d) To get an A in Math 101, it is necessary for you to get an A in the final.

4(a) Express the argument given below using a symbol for each proposition. [10]
(b) Using the symbols of (a) for the propositions, explain how the reasoning proceeds i.e. identify each application of modus ponens in the argument. Is the reasoning correct ? Give your reasons. [16]

Argument:
If today is Sunday, then I do Web surfing or yard work. If the network is down, I can't do Web surfing. It rains only if I can't do yard work. Today is Sunday and it is raining. Therefore I do yard work.