91影视

Use rules of inference to show that the hypotheses 鈥淩andy works hard,鈥� 鈥淚f Randy works hard, then he is a dull boy,鈥� and 鈥淚f Randy is a dull boy, then he will not get the job鈥� imply the conclusion 鈥淩andy will not get the job.鈥�

Let \(P(x)\) be the statement "x spends more than five hours every weekday in class," where the domain for \(x\) consists of all students. Express each of these quantifications in English. $$ \begin{array}{ll}{\text { a) } \quad \exists x P(x)} & {\text { b) } \forall x P(x)} \\ {\text { c) } \quad \exists x \neg P(x)} & {\text { d) } \forall x \neg P(x)}\end{array} $$

Prove that if \(x\) and \(y\) are real numbers, then max \((x, y)+\) \(\min (x, y)=x+y .[\text { Hint: Use a proof by cases, with the }\) two cases corresponding to \(x \geq y\) and \(x

What is the negation of each of these propositions? a) Mei has an MP3 player. b) There is no pollution in New Jersey. c) \(2+1=3\) . d) The summer in Maine is hot and sunny.

Use a truth table to verify the distributive law \(p \wedge(q \vee r) \equiv(p \wedge q) \vee(p \wedge r)\)

In Exercises 1鈥�6, translate the given statement into propositional logic using the propositions provided. You are eligible to be President of the U.S.A. only if you are at least 35 years old, were born in the U.S.A., or at the time of your birth both of your parents were citizens, and you have lived at least 14 years in the country. Express your answer in terms of e: 鈥淵ou are eligible to be President of the U.S.A.,鈥� a: 鈥淵ou are at least 35 years old,鈥� b: 鈥淵ou were born in the U.S.A.,鈥� p: 鈥淎t the time of your birth, both of your parents were citizens,鈥� and r: 鈥淵ou have lived at least 14 years in the U.S.A.鈥�

Use a truth table to verify the first De Morgan law \(\neg(p \wedge q) \equiv \neg p \vee \neg q .\)

What is the negation of each of these propositions? a) Jennifer and Teja are friends. b) There are 13 items in a baker鈥檚 dozen. c) Abby sent more than 100 text messages yesterday. d) 121 is a perfect square.

You can upgrade your operating system only if you have a 32-bit processor running at 1 GHz or faster, at least 1 GB RAM, and 16 GB free hard disk space, or a 64-bit processor running at 2 GHz or faster, at least 2 GB RAM, and at least 32 GB free hard disk space. Express your answer in terms of u: 鈥淵ou can upgrade your operating system,鈥� b32: 鈥淵ou have a 32-bit processor,鈥� b64: 鈥淵ou have a 64-bit processor,鈥� g1: 鈥淵our processor runs at 1 GHz or faster,鈥� g2: 鈥淵our processor runs at 2 GHz or faster,鈥� r1: 鈥淵our processor has at least 1 GB RAM,鈥� r2: 鈥淵our processor has at least 2 GB RAM,鈥� h16: 鈥淵ou have at least 16 GB free hard disk space,鈥� and h32: 鈥淵ou have at least 32 GB free hard disk space.鈥�

Use a proof by cases to show that \(\min (a, \min (b, c))=\) \(\min (\min (a, b), c)\) whenever \(a, b,\) and \(c\) are real numbers.