Question

Let P(x) be the statement "x is a clear explanation," ????(x) be the statement "x is satisfactory," ????(x) be the statement "x is an excuse." Let the domain for x consists of all English written text. Express each of the statements below using quantifiers, the predicates P(x), ????(x), ???????????? ????(x) and logical connectives, including negations if needed. a. Some clear explanations are excuses. b. There are no satisfactory excuses. [Note this is equivalent to: All excuses are not satisfactory.]

Answer 1

a P(x) = Q(x)

b. R(x) = Q(x)

c. R(x) = P(x)

d. Yes

Explanation:

This is a statement of logical connectives.

a P(x) = Q(x)

b. R(x) = Q(x)

c. R(x) = P(x)

d. Yes, (c) follows from (a) and (b)

Reasoning:

(c) is equivalent to ∀x ¬ P(x) ∨ Q(x)

Proof:

Follow the tautology (X→Υ) Ξ(¬x∨Υ)

This gives X: R(x) ∧ ¬ P(X)