Let the universe consists of all people. Consider the following hypotheses: • Some students like all teachers. • No student likes any dishonest person. Using first-order logic prove that "no teacher is dishonest". Use the following predicates for your proof: S(x) denotes x is a student; T(x) denotes x is a teacher; D(x) denotes x is dishonest; L(x,y) denotes x likes y.

Using truth tables, or in any other way, prove that each of the following compound propositions is not a tautology. These implications are common logical fallacies (errors in reasoning) since the conclusion does not follow logically from the set of hypotheses. a. [(P⇒ Q) ^ Q] ⇒ P. b. [(PQ) ^ (~ P)] ⇒ (~ Q). For each one of the logical fallacies in part (iii) of this question give an example of a "real life" situation where such an error can occur.

Evaluate the proposition ∀∃y.P(x,y) ⇐⇒ ∃y.∀x.P(x,y) for an unknown predicate P. Could the statement be true? Must the statement be true for all P? Give a specific P (with domains for x and y) where it is either true or false. Prove that it is true for all P, or show a contradiction.

How can we make a truth table ?