Recent questions tagged kenneth-rosen / GATE Overflow for GATE CSE

Show that these statements are inconsistent: “If Miranda does not take a course in discrete mathematics, then she will not graduate.” “If Miranda does not graduate,...

aditi19

2.6k views

aditi19 asked Mar 26, 2019

0 votes

1 answer

732

Kenneth Rosen Edition 7 Exercise 1.7 Question 7 (Page No. 91)
Use a direct proof to show that every odd integer is the difference of two squares.

Use a direct proof to show that every odd integer is the difference of two squares.

Pooja Khatri

897 views

Pooja Khatri asked Mar 20, 2019

0 votes

0 answers

733

Kenneth Rosen Edition 7 Exercise 1.7 Question 6 (Page No. 91)
Use a direct proof to show that the product of two odd numbers is odd.

Use a direct proof to show that the product of two odd numbers is odd.

Pooja Khatri

211 views

Pooja Khatri asked Mar 20, 2019

1 votes

1 answer

734

Kenneth Rosen Edition 7 Exercise 1.7 Question 5 (Page No. 91)
Prove that if $m+n$ and $n+p$ are even integers, where $m, n$,and $p$ are integers, then $m+p$ is even. What kind of proof did you use?

Prove that if $m+n$ and $n+p$ are even integers, where $m, n$,and $p$ are integers, then $m+p$ is even. What kind of proof did you use?

Pooja Khatri

235 views

Pooja Khatri asked Mar 20, 2019

0 votes

0 answers

735

Kenneth Rosen Edition 7 Exercise 1.7 Question 4 (Page No. 91)
Show that the additive inverse, or negative, of an even number is an even number using a direct proof.

Show that the additive inverse, or negative, of an even number is an even number using a direct proof.

Pooja Khatri

289 views

Pooja Khatri asked Mar 20, 2019

0 votes

1 answer

736

Kenneth Rosen Edition 7 Exercise 1.7 Question 3 (Page No. 91)
Show that the square of an even number is an even number using a direct proof

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

Pooja Khatri

270 views

Pooja Khatri asked Mar 20, 2019

0 votes

0 answers

737

Kenneth Rosen Edition 7 Exercise 1.7 Question 2 (Page No. 91)
Use a direct proof to show that the sum of two even integers is even.

Use a direct proof to show that the sum of two even integers is even.

Pooja Khatri

193 views

Pooja Khatri asked Mar 20, 2019

1 votes

1 answer

738

Kenneth Rosen Edition 7 Exercise 1.7 Question 1 (Page No. 91)
Use a direct proof to show that the sum of two odd integers is even.

Use a direct proof to show that the sum of two odd integers is even.

Pooja Khatri

368 views

Pooja Khatri asked Mar 20, 2019

0 votes

1 answer

739

Kenneth Rosen Edition 7 Exercise 1.6 Question 35 (Page No. 80)
Determine whether this argument, taken from Kalish and Montague [KaMo64], is valid. If Superman were able and willing to prevent evil,he would do so. If Superman were unable to prevent evil, he would be impotent; if ... does not prevent evil. If Superman exists, he is neither impotent nor malevolent. Therefore, Superman does not exist.

Determine whether this argument, taken from Kalish and Montague [KaMo64], is valid.If Superman were able and willing to prevent evil,he would do so. If Superman were unab...

Pooja Khatri

1.2k views

Pooja Khatri asked Mar 20, 2019

0 votes

1 answer

740

Kenneth Rosen Edition 7 Exercise 1.6 Question 34 (Page No. 80)
The Logic Problem, taken from WFF'N PROOF, The Game of Logic, has these two assumptions:1. Logic is difficult or not many students like logic. 2. If mathematics is easy, then logic is not difficult. By translating these ... not easy. That if not many students like logic, then either mathematics is not easy or logic is not difficult.

The Logic Problem, taken from WFF’N PROOF, The Game of Logic, has these two assumptions:1. “Logic is difficult or not many students like logic.”2. “If mathematics...

Pooja Khatri

882 views

Pooja Khatri asked Mar 20, 2019

0 votes

1 answer

741

Kenneth Rosen Edition 7 Exercise 1.6 Question 33 (Page No. 80)
Use resolution to show that the compound proposition $(p \vee q) \wedge (\sim p \vee q) \wedge (p \vee \sim q) \wedge (\sim p \vee \sim q)$ is not satisfiable.

Use resolution to show that the compound proposition $(p \vee q) \wedge (\sim p \vee q) \wedge (p \vee \sim q) \wedge (\sim p \vee \sim q)$ is not satisfiable.

Pooja Khatri

745 views

Pooja Khatri asked Mar 20, 2019

0 votes

0 answers

742

Kenneth Rosen Edition 7 Exercise 1.6 Question 32 (Page No. 80)
Show that the equivalence $p \wedge \sim p \equiv F$ can be derived using resolution together with the fact that a conditional statement with a false hypothesis is true. [Hint:Let $q=r=F$ in resolution.]

Show that the equivalence $p \wedge \sim p \equiv F$ can be derived using resolution together with the fact that a conditional statement with a false hypothesis is true. ...

Pooja Khatri

346 views

Pooja Khatri asked Mar 20, 2019

1 votes

1 answer

743

Kenneth Rosen Edition 7 Exercise 1.6 Question 31 (Page No. 80)
Use resolution to show that the hypotheses “It is not raining or Yvette has her umbrella,” “Yvette does not have her umbrella or she does not get wet,” and “It is raining or Yvette does not get wet” imply that “Yvette does not get wet.”

Use resolution to show that the hypotheses “It is not raining or Yvette has her umbrella,” “Yvette does not have her umbrella or she does not get wet,” and “It ...

Pooja Khatri

1.5k views

Pooja Khatri asked Mar 20, 2019

0 votes

1 answer

744

Kenneth Rosen Edition 7 Exercise 1.6 Question 30 (Page No. 80)
Use resolution to show the hypotheses “Allen is a bad boy or Hillary is a good girl” and “Allen is a good boy or David is happy” imply the conclusion “Hillary is a good girl or David is happy.”

Use resolution to show the hypotheses “Allen is a bad boy or Hillary is a good girl” and “Allen is a good boy or David is happy” imply the conclusion “Hillary i...

Pooja Khatri

1.8k views

Pooja Khatri asked Mar 20, 2019

0 votes

0 answers

745

Kenneth Rosen Edition 7 Exercise 1.6 Question 29 (Page No. 80)
Use rules of inference to show that if $\forall x (P(x) \vee Q(x))$, $\forall x (\sim Q(x) \vee S(x)), \forall x (R(x) \rightarrow \sim S(x)),$ and $\exists x \sim P(x)$ are true, then $\exists x \sim R(x)$ is true.

Use rules of inference to show that if $\forall x (P(x) \vee Q(x))$, $\forall x (\sim Q(x) \vee S(x)), \forall x (R(x) \rightarrow \sim S(x)),$ and $\exists x \sim P(x)$ ...

Pooja Khatri

281 views

Pooja Khatri asked Mar 20, 2019

3 votes

1 answer

746

Kenneth Rosen Edition 7 Exercise 1.6 Question 28 (Page No. 80)
Use rules of inference to show that if $\forall x (P(x) \vee Q(x))$ and $\forall x ((\sim P(x) \wedge Q(x)) \rightarrow R(x))$ are true, then $\forall x (\sim R(x) \rightarrow P(x))$ is also true, where the domains of all quantifiers are the same.

Use rules of inference to show that if $\forall x (P(x) \vee Q(x))$ and $\forall x ((\sim P(x) \wedge Q(x)) \rightarrow R(x))$ are true, then $\forall x (\sim R(x) \right...

Pooja Khatri

444 views

Pooja Khatri asked Mar 19, 2019

0 votes

0 answers

747

Kenneth Rosen Edition 7 Exercise 1.6 Question 27 (Page No. 80)
Use rules of inference to show that if $\forall x (P(x) \rightarrow (Q(x) \wedge S(x)))$ and $\forall x ( P(x) \wedge R(x))$ are true, then $\forall x (R(x) \wedge S(x))$ is true.

Use rules of inference to show that if $\forall x (P(x) \rightarrow (Q(x) \wedge S(x)))$ and $\forall x ( P(x) \wedge R(x))$ are true, then $\forall x (R(x) \wedge S(x))$...

Pooja Khatri

219 views

Pooja Khatri asked Mar 19, 2019

0 votes

0 answers

748

Kenneth Rosen Edition 7 Exercise 1.6 Question 26 (Page No. 80)
Justify the rule of universal transitivity, which states that if $\forall x (P(x) \rightarrow Q(x))$ and $\forall x(Q(x) \rightarrow R(x))$ are true, then $\forall x (P(x) \rightarrow R(x))$ is true, where the domains of all quantifiers are the same.

Justify the rule of universal transitivity, which states that if $\forall x (P(x) \rightarrow Q(x))$ and $\forall x(Q(x) \rightarrow R(x))$ are true, then $\forall x (P(x...

Pooja Khatri

220 views

Pooja Khatri asked Mar 19, 2019

0 votes

0 answers

749

Kenneth Rosen Edition 7 Exercise 1.6 Question 25 (Page No. 80)
Justify the rule of universal modus tollens by showing that the premises $ \forall x (P(x) \rightarrow Q(x)) $ and $\sim Q(a)$for $a$ particular element $a$ in the domain, imply $\sim P(a)$

Justify the rule of universal modus tollens by showing that the premises $ \forall x (P(x) \rightarrow Q(x)) $ and $\sim Q(a)$for $a$ particular element $a$ in the domain...

Pooja Khatri

294 views

Pooja Khatri asked Mar 19, 2019

0 votes

0 answers

750

Kenneth Rosen Edition 7 Exercise 1.6 Question 18 (Page No. 79)
What is wrong with this argument? Let $S(x, y)$ be “$x$ is shorter than $y$.” Given the premise $\exists s S(s, Max)$, it follows that $S(Max, Max)$. Then by existential generalization it follows that $\exists x S(x,x)$, so that someone is shorter than himself.

What is wrong with this argument? Let $S(x, y)$ be “$x$ is shorter than $y$.” Given the premise $\exists s S(s, Max)$, it follows that $S(Max, Max)$. Then by existent...

Pooja Khatri

465 views

Pooja Khatri asked Mar 19, 2019

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