Recent questions tagged discrete-mathematics / GATE Overflow for GATE CSE

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1051

Kenneth Rosen Edition 7 Exercise 1.5 Question 20 (Page No. 66)
Express each of these statements using predicates, quantifiers, logical connectives, and mathematical operators where the domain consists of all integers. The product of two negative integers is positive. The average of two positive integers ... of the sum of two integers does not exceed the sum of the absolute values of these integers.

Express each of these statements using predicates, quantifiers, logical connectives, and mathematical operators where the domain consists of all integers.The product of t...

Pooja Khatri

365 views

Pooja Khatri asked Mar 18, 2019

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1052

Kenneth Rosen Edition 7 Exercise 1.5 Question 19 (Page No. 66)
Express each of these statements using mathematical and logical operators, predicates, and quantifiers, where the domain consists of all integers. The sum of two negative integers is negative. The difference of two positive integers ... sum. The absolute value of the product of two integers is the product of their absolute values.

Express each of these statements using mathematical and logical operators, predicates, and quantifiers, where the domain consists of all integers.The sum of two negative ...

Pooja Khatri

404 views

Pooja Khatri asked Mar 18, 2019

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1053

Kenneth Rosen Edition 7 Exercise 1.5 Question 18 (Page No. 66)
Express each of these system specifications using predicates, quantifiers, and logical connectives, if necessary. At least one console must be accessible during every fault condition. The e-mail address of every user can be ... knows the password of every user on the system except for the system administrator, who knows all passwords.

Express each of these system specifications using predicates, quantifiers, and logical connectives, if necessary.At least one console must be accessible during every faul...

Pooja Khatri

384 views

Pooja Khatri asked Mar 18, 2019

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1054

Kenneth Rosen Edition 7 Exercise 1.5 Question 17 (Page No. 66)
Express each of these system specifications using predicates, quantifiers, and logical connectives, if necessary. Every user has access to exactly one mailbox. There is a process that continues to run during all error conditions only ... whose url has a .edu extension. There are exactly two systems that monitor every re-mote server.

Express each of these system specifications using predicates, quantifiers, and logical connectives, if necessary.Every user has access to exactly one mailbox.There is a p...

Pooja Khatri

302 views

Pooja Khatri asked Mar 18, 2019

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1055

Kenneth Rosen Edition 7 Exercise 1.5 Question 16 (Page No. 66)
A discrete mathematics class contains 1 mathematics ma-jor who is a freshman, 12 mathematics majors who are sophomores, 15 computer science majors who are sophomores, 2 mathematics majors who are juniors, 2 computer science majors who are ... is a major such that there is a student in the class in every year of study with that major

A discrete mathematics class contains 1 mathematics ma-jor who is a freshman, 12 mathematics majors who are sophomores, 15 computer science majors who are sophomores, 2 m...

Pooja Khatri

438 views

Pooja Khatri asked Mar 18, 2019

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1056

Kenneth Rosen Edition 7 Exercise 1.5 Question 15 (Page No. 66)
Use quantifiers and predicates with more than one variable to express these statements. Every computer science student needs a course in discrete mathematics There is a student in this class who owns a personal computer. Every student in ... campus. Every student in this class has been in at least one room of every building on campus.

Use quantifiers and predicates with more than one variable to express these statements.Every computer science student needs a course in discrete mathematicsThere is a stu...

Pooja Khatri

875 views

Pooja Khatri asked Mar 18, 2019

1 votes

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1057

Kenneth Rosen Edition 7 Exercise 1.5 Question 13 (Page No. 66)
Let $M(x, y)$ be $x$ has sent $y$ an e-mail message and $T (x, y)$ be $x$ has telephoned $y$, where the domain consists of all students in your class. Use quantifiers to express each of these ... There are two different students in your class who between them have sent an e-mail message to or telephoned everyone else in the class.

Let $M(x, y)$ be “$x$ has sent $y$ an e-mail message” and $T (x, y)$ be “$x$ has telephoned $y$,” where the domain consists of all students in your class. Use qua...

Pooja Khatri

1.2k views

Pooja Khatri asked Mar 18, 2019

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1058

Kenneth Rosen Edition 7 Exercise 1.5 Question 12 (Page No. 65)
Let $I(x)$ be the statement $x$ has an Internet connection and $C(x, y)$ be the statement $x$ and $y$ have chatted over the Internet, where the domain for the variables $x$ and $y$ consists of all ... the same person in your class. There are two students in the class who between them have chatted with everyone else in the class.

Let $I(x)$ be the statement “$x$ has an Internet connection”and $C(x, y)$ be the statement “$x$ and $y$ have chatted over the Internet,” where the domain for the ...

Pooja Khatri

468 views

Pooja Khatri asked Mar 18, 2019

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1059

Kenneth Rosen Edition 7 Exercise 1.5 Question 11 (Page No. 65)
Let$S(x)$ be the predicate $x$ is a student, $F(x)$ the predicate $x$ is a faculty member, and $A(x, y)$ the predicate $x$ has asked $y$ a question, where the domain consists of all people ... is a faculty member who has asked every other faculty member a question. Some student has never been asked a question by a faculty member

Let$S(x)$ be the predicate “$x$ is a student,”$F(x)$ the predicate “$x$ is a faculty member,” and $A(x, y)$ the predicate“$x$ has asked $y$ a question,” where...

Pooja Khatri

494 views

Pooja Khatri asked Mar 18, 2019

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1060

Kenneth Rosen Edition 7 Exercise 1.5 Question 10 (Page No. 65)
Let $F(x, y)$ be the statement $x$ can fooly, where the domain consists of all people in the world. Use quantifiers to express each of these statements. Everybody can fool Fred. Evelyn can fool everybody.Everybody can ... . No one can fool himself or herself. There is someone who can fool exactly one person besides himself or herself.

Let $F(x, y)$ be the statement “$x$ can fooly,” where the domain consists of all people in the world. Use quantifiers to express each of these statements.Everybody ca...

Pooja Khatri

403 views

Pooja Khatri asked Mar 18, 2019

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1061

Kenneth Rosen Edition 7 Exercise 1.5 Question 9 (Page No. 65)
Let $L(x, y)$ be the statement $x$ loves $y$, where the domain for both $x$ and $y$ consists of all people in the world. Use quantifiers to express each of these statements. Everybody loves Jerry. Everybody ... two people whom Lynn loves. Everyone loves himself or herself. There is someone who loves no one besides himself or herself.

Let $L(x, y)$ be the statement “$x$ loves $y$,” where the domain for both $x$ and $y$ consists of all people in the world. Use quantifiers to express each of these st...

Pooja Khatri

591 views

Pooja Khatri asked Mar 18, 2019

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1062

Kenneth Rosen Edition 7 Exercise 1.5 Question 4 (Page No. 64)
Let $P(x, y)$ be the statement Student $x$ has taken classy, where the domain for $x$ consists of all students in your class and for $y$ ... $\forall y \exists x P(x,y)$ $\forall x \forall y P(x,y)$

Let $P(x, y)$ be the statement “Student $x$ has taken classy,” where the domain for $x$ consists of all students in your class and for $y$ consists of all computer sc...

Pooja Khatri

341 views

Pooja Khatri asked Mar 18, 2019

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1063

Kenneth Rosen Edition 7 Exercise 1.5 Question 3 (Page No. 64)
Let $Q(x, y)$ be the statement $x$ has sent an e-mail message toy, where the domain for both $x$ and $y$ consists of all students in your class. Express each of these quantifications in English. $\exists x \exists y Q(x,y)$ ... $\exists y \forall x Q(x,y)$ $\forall y \exists x Q(x,y)$ $\forall x \forall y Q(x,y)$

Let $Q(x, y)$ be the statement “$x$ has sent an e-mail message toy,” where the domain for both $x$ and $y$ consists of all students in your class. Express each of the...

Pooja Khatri

293 views

Pooja Khatri asked Mar 18, 2019

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1064

Kenneth Rosen Edition 7 Exercise 1.5 Question 2 (Page No. 64)
Translate these statements into English, where the domain for each variable consists of all real numbers. $\exists x \forall y (xy =y) $ $ \forall x \forall y (((x>=0) \wedge (y<0)) \rightarrow(x-y>0)) $ $\forall x \forall y \exists z (x=y+z)$

Translate these statements into English, where the domain for each variable consists of all real numbers.$\exists x \forall y (xy =y) $$ \forall x \forall y (((x>=0) \wed...

Pooja Khatri

350 views

Pooja Khatri asked Mar 18, 2019

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1065

Kenneth Rosen Edition 7 Exercise 1.5 Question 1 (Page No. 64)
Translate these statements into English, Where the domain for each variable consists of all real numbers. $\forall x \exists y(x<y) $ $ \forall x \forall y((x>=0) \wedge (y>=0) \rightarrow (xy>=0)) $ $\forall x \forall y \exists z (xy=z)$

Translate these statements into English, Where the domain for each variable consists of all real numbers.$\forall x \exists y(x<y) $$ \forall x \forall y((x>=0) \wedge (y...

Pooja Khatri

597 views

Pooja Khatri asked Mar 18, 2019

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1066

Kenneth Rosen Edition 7 Exercise 1.4 Question 62 (Page No. 56)
Let $P(x),Q(x),R(x)$, and $S(x)$ be the statements $x$ is a duck, $x$ is one of my poultry, $x$ is an officer, and $x$ is willing to waltz, respectively. Express each of these statements using quantifiers; ... ducks. My poultry are not officers. Does (d) follow from (a), (b), and (c)? If not, is there a correct conclusion?

Let $P(x),Q(x),R(x)$, and $S(x)$ be the statements “$x$ is a duck,” “$x$ is one of my poultry,” “$x$ is an officer,”and “$x$ is willing to waltz,” respect...

Pooja Khatri

578 views

Pooja Khatri asked Mar 18, 2019

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1067

Kenneth Rosen Edition 7 Exercise 1.4 Question 60 (Page No. 56)
Let$P(x),Q(x),R(x)$, and $S(x)$ be the statements $x$ is a baby, $x$ is logical, $x$ is able to manage a crocodile, and $x$is despised, respectively. Suppose that the domain consists of all people. Express ... despised. Babies cannot manage crocodiles. Does (d) follow from (a), (b), and (c)? If not, is there a correct conclusion

Let$P(x),Q(x),R(x)$, and $S(x)$ be the statements “$x$ is a baby,” “$x$ is logical,” “$x$ is able to manage a crocodile,”and “$x$is despised,” respectivel...

Pooja Khatri

402 views

Pooja Khatri asked Mar 18, 2019

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1068

Kenneth Rosen Edition 7 Exercise 1.4 Question 59 (Page No. 56)
Let $P(x),Q(x),$ and $R(x)$ be the statements $x$ is a professor, $x$ is ignorant, and $x$ is vain, respectively.Express each of these statements using quantifiers; logical connectives; and $P(x),Q(x)$, ... people. No professors are ignorant. All ignorant people are vain. No professors are vain. Does (c) follow from (a) and (b)?

Let $P(x),Q(x),$ and $R(x)$ be the statements “$x$ is a professor,” “$x$ is ignorant,” and “$x$ is vain,” respectively.Express each of these statements using ...

Pooja Khatri

369 views

Pooja Khatri asked Mar 18, 2019

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1069

Kenneth Rosen Edition 7 Exercise 1.4 Question 58 (Page No. 56)
Suppose that Prolog facts are used to define the predicates mother $(M,Y)$ and father (F,X), which represent that M is the mother of $Y$ and $F$ is the father of $X$ respectively. Give a Prolog rule to define the predicate grandfather $(X,Y)$, which represent that $X$ is the grandfather of $Y$.

Suppose that Prolog facts are used to define the predicates mother $(M,Y)$ and father (F,X), which represent that M is the mother of $Y$ and $F$ is the father of $X$ resp...

Pooja Khatri

312 views

Pooja Khatri asked Mar 18, 2019

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1070

Kenneth Rosen Edition 7 Exercise 1.4 Question 57 (Page No. 56)
Suppose that Prolog facts are used to define the predicates mother $(M,Y)$ and father $(F,X)$, which represent that $M$ is the mother of $Y$ and $F$ is the father of $X$, respectively. Give a Prolog rule to define the predicate ... $(X,Y)$ , which represent that $X$, $Y$ are siblings (that is, have the same mother and the same father.)

Suppose that Prolog facts are used to define the predicates mother $(M,Y)$ and father $(F,X)$, which represent that $M$ is the mother of $Y$ and $F$ is the father of $X$,...

Pooja Khatri

266 views

Pooja Khatri asked Mar 18, 2019

1 votes

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1071

Kenneth Rosen Edition 7 Exercise 1.4 Question 53 (Page No. 56)
What are the truth values of these statements? $\exists \sim x P(x) \rightarrow \exists x P(x) $ $\forall x P(x) \rightarrow \exists \sim x P(x) $ $\exists \sim x \sim P(x) \rightarrow \sim \forall x P(x) $

What are the truth values of these statements?$\exists \sim x P(x) \rightarrow \exists x P(x) $$\forall x P(x) \rightarrow \exists \sim x P(x) $$\exists \sim x \sim P(x)...

Pooja Khatri

319 views

Pooja Khatri asked Mar 18, 2019

2 votes

1 answer

1072

Kenneth Rosen Edition 7 Exercise 1.4 Question 52 (Page No. 56)
As mentioned in the text, the notation$\exists \sim xP (x)$ denotes “There exists a unique $x$ such that $P(x)$ is true.”If the domain consists of all integers, what are the truth values of these statements? $\exists \sim x(x>1)$ $\exists \sim x (x^2 = 1)$ $\exists x (x+3 = 2x)$ $\exists \sim x(x = x+1)$

As mentioned in the text, the notation$\exists \sim xP (x)$ denotes “There exists a unique $x$ such that $P(x)$ is true.”If the domain consists of all integers, what ...

Pooja Khatri

560 views

Pooja Khatri asked Mar 18, 2019

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1 answer

1073

Kenneth Rosen Edition 7 Exercise 1.4 Question 50 (Page No. 56)
Show that $\forall x P(x) \vee \forall x Q(x)$ and $\forall x (P(x) \vee Q(x))$ are not logically equivalent.

Show that $\forall x P(x) \vee \forall x Q(x)$ and $\forall x (P(x) \vee Q(x))$ are not logically equivalent.

Pooja Khatri

297 views

Pooja Khatri asked Mar 18, 2019

2 votes

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1074

Kenneth Rosen Edition 7 Exercise 1.4 Question 51 (Page No. 56)
Show that $\exists x P(x) \wedge \exists x Q(x)$ and $\exists x (P(x) \wedge Q(x))$ are not logically equivalent.

Show that $\exists x P(x) \wedge \exists x Q(x)$ and $\exists x (P(x) \wedge Q(x))$ are not logically equivalent.

Pooja Khatri

240 views

Pooja Khatri asked Mar 18, 2019

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1075

Kenneth Rosen Edition 7 Exercise 1.4 Question 49 (Page No. 56)
Establish these logical equivalences, where $x$ does not occur as a free variable in $A$. Assume that the domain is nonempty. $\forall x P(x) \rightarrow A \equiv \exists x P(x) \rightarrow A$ $\exists x P(x) \rightarrow A \equiv \forall x (P(x) \rightarrow A$

Establish these logical equivalences, where $x$ does not occur as a free variable in $A$. Assume that the domain is nonempty.$\forall x P(x) \rightarrow A \equiv \exists ...

Pooja Khatri

186 views

Pooja Khatri asked Mar 18, 2019

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1 answer

1076

Kenneth Rosen Edition 7 Exercise 1.4 Question 47 (Page No. 56)
Establish these logical equivalences, where $x$ does not occur as a free variable in $A$. Assume that the domain is nonempty. $(\forall x P(x)) \wedge A \equiv \forall x (P(x) \wedge A)$ $(\exists x P(x)) \wedge A \equiv \exists x (P(x) \wedge A)$

Establish these logical equivalences, where $x$ does not occur as a free variable in $A$. Assume that the domain is nonempty.$(\forall x P(x)) \wedge A \equiv \forall x (...

Pooja Khatri

405 views

Pooja Khatri asked Mar 18, 2019

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1077

Kenneth Rosen Edition 7 Exercise 1.4 Question 46 (Page No. 56)
Establish these logical equivalences, where $x$ does not occur as a free variable in $A$. Assume that the domain is nonempty. $(\forall x P(x)) \vee A \equiv \forall x (P(x) \vee A)$ $(\exists x P(x)) \vee A \equiv \exists x (P(x) \vee A)$

Establish these logical equivalences, where $x$ does not occur as a free variable in $A$. Assume that the domain is nonempty.$(\forall x P(x)) \vee A \equiv \forall x (P(...

Pooja Khatri

284 views

Pooja Khatri asked Mar 18, 2019

1 votes

1 answer

1078

Kenneth Rosen Edition 7 Exercise 1.4 Question 45 (Page No. 56)
Show that $\exists x (P(x) \vee Q(x))$ and $\exists x P(x) \vee \exists x Q(x)$ are logically equivalent.

Show that $\exists x (P(x) \vee Q(x))$ and $\exists x P(x) \vee \exists x Q(x)$ are logically equivalent.

Pooja Khatri

314 views

Pooja Khatri asked Mar 18, 2019

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1079

Kenneth Rosen Edition 7 Exercise 1.4 Question 44 (Page No. 56)
Determine whether $\forall x (P(x) \leftrightarrow Q(x))$ and $\forall x P(x) \leftrightarrow \forall xQ(x)$ are logically equivalent . Justify your answer.

Determine whether $\forall x (P(x) \leftrightarrow Q(x))$ and $\forall x P(x) \leftrightarrow \forall xQ(x)$ are logically equivalent . Justify your answer.

Pooja Khatri

206 views

Pooja Khatri asked Mar 18, 2019

2 votes

2 answers

1080

Kenneth Rosen Edition 7 Exercise 1.4 Question 43 (Page No. 56)
Determine whether $\forall x (P(x) \rightarrow Q(x))$ and $\forall x P(x) \rightarrow \forall xQ(x)$ are logically equivalent . Justify your answer.

Determine whether $\forall x (P(x) \rightarrow Q(x))$ and $\forall x P(x) \rightarrow \forall xQ(x)$ are logically equivalent . Justify your answer.

Pooja Khatri

990 views

Pooja Khatri asked Mar 18, 2019

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force match	+apple
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