91影视

In Exercises 1鈥�6, translate the given statement into propositional logic using the propositions provided. You can graduate only if you have completed the requirements of your major and you do not owe money to the university and you do not have an overdue library book. Express your answer in terms of g: 鈥淵ou can graduate,鈥� m: 鈥淵ou owe money to the university,鈥� r: 鈥淵ou have completed the requirements of your major,鈥� and b: 鈥淵ou have an overdue library book.鈥�

Use a proof by cases to show that 100 is not the cube of a positive integer. [Hint: Consider two cases: (i) \(1 \leq x \leq 4\) , (ii) \(x \geq 5 . ]\)

Show that the square of an even number is an even number using a direct proof.

What is the negation of each of these propositions? a) Linda is younger than Sanjay. b) Mei makes more money than Isabella. c) Moshe is taller than Monica. d) Abby is richer than Ricardo.

Let \(Q(x, y)\) be the statement "x has sent an e-mail message to \(y,\) " where the domain for both \(x\) and \(y\) consists of all students in your class. Express each of these quantifications in English. $$ \begin{array}{ll}{\text { a) } \exists x \exists y Q(x, y)} & {\text { b) } \exists x \forall y Q(x, y)} \\ {\text { c) } \forall x \exists y Q(x, y)} & {\text { d) } \exists y \forall x Q(x, y)} \\ {\text { e) } \forall y \exists x Q(x, y)} & {\text { f) } \forall x \forall y Q(x, y)}\end{array} $$

Let \(P(x, y)\) be the statement "Student \(x\) has taken class \(y,\) where the domain for \(x\) consists of all students in your class and for \(y\) consists of all computer science courses at your school. Express each of these quantifications in English. $$ \begin{array}{ll}{\text { a) } \exists x \exists y P(x, y)} & {\text { b) } \exists x \forall y P(x, y)} \\ {\text { c) } \forall x \exists y P(x, y)} & {\text { d) } \exists y \forall x P(x, y)} \\ {\text { e) } \forall y \exists x P(x, y)} & {\text { f) } \forall x \forall y P(x, y)}\end{array} $$

What rule of inference is used in each of these arguments? a) Kangaroos live in Australia and are marsupials. Therefore, kangaroos are marsupials. b) It is either hotter than 100 degrees today or the pollution is dangerous. It is less than 100 degrees outside today. Therefore, the pollution is dangerous. c) Linda is an excellent swimmer. If Linda is an excellent swimmer, then she can work as a lifeguard. Therefore, Linda can work as a lifeguard. d) Steve will work at a computer company this summer. Therefore, this summer Steve will work at a computer company or he will be a beach bum. e) If I work all night on this homework, then I can answer all the exercises. If I answer all the exercises, I will understand the material. Therefore, if I work all night on this homework, then I will understand the material.

Prove that there are no positive perfect cubes less than 1000 that are the sum of the cubes of two positive integers.

What is the negation of each of these propositions? a) Janice has more Facebook friends than Juan. b) Quincy is smarter than Venkat. c) Zelda drives more miles to school than Paola. d) Briana sleeps longer than Gloria.

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�