Question

3. Suppose that the domain for x consists of all English text.

P(x):“x is a clear explanation,”

Q(x): “x is satisfactory,” and

R(x):“x is an excuse,”

Express each of these statements using quantifiers, logical connectives, and P (x), Q(x),and R(x).

a) Some clear explanations are satisfactory

b) All excuses are unsatisfactory

c) Some excuses are not clear explanations.

d) Does (c) follow from (a) and (b)?

4. Prove that if you pick four utensils from a drawer containing just spoons, forks, and knives, you must get either a pair of forks or a pair of spoons or a pair of knives.

5. Prove that there is a positive integer that equals the sum of the positive integers not exceeding it. Is your proof constructive or non-constructive?

6. Prove that 4n < ?! if n is an integer greater than 8.

Answer 1

Suppose that the domain for x consists of all English text.

P(x):“x is a clear explanation,”

Q(x): “x is satisfactory,” and

R(x):“x is an excuse,”

a) Some clear explanations are satisfactory

b) All excuses are unsatisfactory

c) Some excuses are not clear explanations.

d) Does (c) follow from (a) and (b)

Yes, (c) follow from (a) that shows some not clear explanations are satisfactory.