91Ó°ÊÓ

For each pair of propositions \(P\) and \(Q\) . State whether or not \(P \equiv Q\). $$ P=p, Q=p \vee q $$

Refer to a group of 191 students, of which 10 are taking French, business, and music; 36 are taking French and business; 20 are taking French and music; 18 are taking business and music; 65 are taking French; 76 are taking business; and 63 are taking music. How many are taking French or business (or both)?

A television poll of 151 persons found that 68 watched "Law and Disorder"; 61 watched "25"; 52 watched "The Tenors"; 16 watched both "Law and Disorder" and "25"; 25 watched both "Law and Disorder" and "The Tenors"; 19 watched both "25" and "The Tenors"; and 26 watched none of these shows. How many persons watched all three shows?

For each pair of propositions \(P\) and \(Q\) . State whether or not \(P \equiv Q\). $$ P=(p \rightarrow q) \wedge(q \rightarrow r), Q=p \rightarrow r $$

For each pair of propositions \(P\) and \(Q\) . State whether or not \(P \equiv Q\). $$ P=(s \rightarrow(p \wedge \neg r)) \wedge((p \rightarrow(r \vee q)) \wedge s), Q=p \vee t $$

Which rule of inference is used in the following argument? Every rational number is of the form \(p / q,\) where \(p\) and \(q\) are integers. Therefore, 9.345 is of the form \(p / q\)

At one time, the following ordinance was in effect in Naperville, Illinois: "It shall be unlawful for any person to keep more than three [3] dogs and three [3] cats upon his property within the city." Was Charles Marko, who owned five dogs and no cats, in violation of the ordinance? Explain.

Verify the second of De Morgan's laws, \(\neg(p \wedge q) \equiv \neg p \vee \neg q\).

Assume that \(\exists x \forall y P(x, y)\) is false and that the domain of discourse is nonempty. Which of must also be false? Prove your answer. $$ \forall x \exists y P(x, y) $$

Assume that \(\exists x \exists y P(x, y)\) is false and that the domain of discourse is nonempty. Which of must also be false? Prove your answer. $$ \forall x \exists y P(x, y) $$