1. Express the assertions given below as a proposition of a predicate logic using the following predicates. The universe is the set of integers.
G(x,y): x > y
N(x): x is a natural number.
E(x): x is even.
(a) There are even numbers among the integers.
(b) Every natural number is negative or even.
(c) An integer is greater than 0 only if it is a natural number.
(d) 2 is even.
(e) For an integer to be a natural number, it must be nonnegative.
2. Fill in the blanks with the shortest string of characters so that the resultant proposition is valid.
(a)
(b)
(c)
(d) [
3. Simplify the following proposition (write all intermediate steps):
(a)
(b)
4. Find the converse and contrapositive for each of the following propositions:
(a) If an argument is wrong, then it is not valid.
(b) For a list of numbers to be sorted, it is necessary that the largest number in the list can be found quickly.
(c) 3 is a prime number only if it is not divisible by 2.