91影视

Suppose the domain of the propositional function \(P(x, y)\) consists of pairs \(x\) and \(y,\) where \(x\) is \(1,2,\) or 3 and \(y\) is \(1,2,\) or \(3 .\) Write out these propositions using disjunctions and conjunctions. $$ \begin{array}{ll}{\text { a) } \exists x P(x, 3)} & {\text { b) } \forall y P(1, y)} \\ {\text { c) } \exists y \neg P(2, y)} & {\text { d) } \forall x \neg P(x, 2)}\end{array} $$

Prove that \(m^{2}=n^{2}\) if and only if \(m=n\) or \(m=-n\)

State the converse, contrapositive, and inverse of each of these conditional statements. a) If it snows tonight, then I will stay at home. b) I go to the beach whenever it is a sunny summer day. c) When I stay up late, it is necessary that I sleep until noon.

Prove that there is no positive integer \(n\) such that \(n^{2}+\) \(n^{3}=100 .\)

Exercises 28鈥�35 relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies (called normals by Smullyan [Sm78]) who can either lie or tell the truth. You encounter three people, A, B, and C. You know one of these people is a knight, one is a knave, and one is a spy. Each of the three people knows the type of person each of other two is. For each of these situations, if possible, determine whether there is a unique solution and determine who the knave, knight, and spy are. When there is no unique solution, list all possible solutions or state that there are no solutions. A says 鈥淚 am the knight,鈥� B says 鈥淎 is telling the truth,鈥� and C says 鈥淚 am the spy.鈥�

Express the negations of each of these statements so that all negation symbols immediately precede predicates. a) \(\forall x \exists y \forall z T(x, y, z)\) b) \(\forall x \exists y P(x, y) \vee \forall x \exists y Q(x, y)\) c) \(\forall x \exists y P(x, y) \wedge \exists z R(x, y, z) )\) d) \(\forall x \exists y(P(x, y) \rightarrow Q(x, y))\)

Prove or disprove that if \(m\) and \(n\) are integers such that \(m n=1,\) then either \(m=1\) and \(n=1,\) or else \(m=-1\) and \(n=-1\)

Suppose that the domain of \(Q(x, y, z)\) consists of triples \(x, y, z,\) where \(x=0,1,\) or \(2, y=0\) or \(1,\) and \(z=0\) or \(1 .\) Write out these propositions using disjunctions and conjunctions. $$ \begin{array}{ll}{\text { a) } \forall y Q(0, y, 0)} & {\text { b) } \exists x Q(x, 1,1)} \\ {\text { c) } \exists z \neg Q(0,0, z)} & {\text { d) } \exists x \neg Q(x, 0,1)}\end{array} $$

Use resolution to show that the hypotheses 鈥淚t is not raining or Yvette has her umbrella,鈥� 鈥淵vette does not have her umbrella or she does not get wet,鈥� and 鈥淚t is raining or Yvette does not get wet鈥� imply that 鈥淵vette does not get wet.鈥�

Each of Exercises \(20-32\) asks you to show that two compound propositions are logically equivalent. To do this, either show that both sides are true, or that both sides are false, for exactly the same combinations of truth values of the propositional variables in these expressions (whichever is easier). Show that \(p \leftrightarrow q\) and \((p \rightarrow q) \wedge(q \rightarrow p)\) are logically equivalent.