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

0 votes

0 answers

691

Kenneth Rosen Edition 7 Exercise 2.1 Question 5 (Page No. 125)
Determine whether each of these pairs of sets are equal. { $1,3,3,3,5,5,5,5,5$ }, { $5,3,1$ } {{ $1$ }}, { $1$ , { $1$ }} $\phi$, { $\phi$ }

Determine whether each of these pairs of sets are equal.{ $1,3,3,3,5,5,5,5,5$ }, { $5,3,1$ }{{ $1$ }}, { $1$ , { $1$ }}$\phi$, { $\phi$ }

Pooja Khatri

323 views

Pooja Khatri asked Apr 5, 2019

0 votes

0 answers

692

Kenneth Rosen Edition 7 Exercise 2.1 Question 4 (Page No. 125)
For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first,or neither is a subset of the other. the set of people who speak English, the ... of fruits, the set of citrus fruits the set of students studying discrete mathematics, the set of students studying data structures

For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first,or neither is a subset of the other.the set of...

Pooja Khatri

337 views

Pooja Khatri asked Apr 5, 2019

0 votes

0 answers

693

Kenneth Rosen Edition 7 Exercise 2.1 Question 3 (Page No. 125)
For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other. the set of airline flights from New York to New Delhi ... English, the set of people who speak Chinese he set of flying squirrels, the set of living creatures that can fly.

For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other.the set o...

Pooja Khatri

346 views

Pooja Khatri asked Apr 5, 2019

1 votes

0 answers

694

Kenneth Rosen Edition 7 Exercise 2.1 Question 2 (Page No. 125)
Use set builder notation to give a description of each of these sets. { $0,3,6,9,12$ } { $-3,-2,-1,0,1,2,3$ } { $ m,n,o,p$ }

Use set builder notation to give a description of each of these sets.{ $0,3,6,9,12$ }{ $-3,-2,-1,0,1,2,3$ }{ $ m,n,o,p$ }

Pooja Khatri

204 views

Pooja Khatri asked Apr 5, 2019

0 votes

0 answers

695

Kenneth Rosen Edition 7 Exercise 2.1 Question 1 (Page No. 125)
List the numbers of these sets. { $x$ | $x$ is a real number such that $x^2 =1$ } { $x$ | $x$ is a positive integer less than 12 } { $x$ | $x$ is the square of an integer and $x<100$ } { $x$ | $x$ is an integer such that $x^2 =2$ }

List the numbers of these sets.{ $x$ | $x$ is a real number such that $x^2 =1$ }{ $x$ | $x$ is a positive integer less than 12 }{ $x$ | $x$ is the square of an integer an...

Pooja Khatri

234 views

Pooja Khatri asked Apr 5, 2019

0 votes

0 answers

696

Kenneth Rosen Edition 7 Exercise 1.7 Question 42 (Page No. 92)
Prove that these four statements about the integer $n$ are equivalent: $n^2$is odd, $1−n$ is even, $n^3$ is odd, $n^2+1$ is even.

Prove that these four statements about the integer $n$ are equivalent:$n^2$is odd,$1−n$ is even,$n^3$ is odd,$n^2+1$ is even.

Pooja Khatri

295 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

697

Kenneth Rosen Edition 7 Exercise 1.7 Question 41 (Page No. 92)
Prove that if $n$ is an integer, these four statements are equivalent: $n$ is even, $n+1$ is odd, $3n+1$isodd, $3n$ is even.

Prove that if $n$ is an integer, these four statements are equivalent:$n$ is even,$n+1$ is odd,$3n+1$isodd,$3n$ is even.

Pooja Khatri

245 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

698

Kenneth Rosen Edition 7 Exercise 1.7 Question 39 (Page No. 92)
Prove that at least one of the real numbers $a_1,a_2,...,a_n$ is greater than or equal to the average of these numbers.What kind of proof did you use?

Prove that at least one of the real numbers $a_1,a_2,...,a_n$ is greater than or equal to the average of these numbers.What kind of proof did you use?

Pooja Khatri

209 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

699

Kenneth Rosen Edition 7 Exercise 1.7 Question 38 (Page No. 92)
Find a counterexample to the statement that every positive integer can be written as the sum of the squares of three integers

Find a counterexample to the statement that every positive integer can be written as the sum of the squares of three integers

Pooja Khatri

329 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

700

Kenneth Rosen Edition 7 Exercise 1.7 Question 37 (Page No. 91)
Show that the propositions $p1,p2,p3,p4,$ and $p5$ can be shown to be equivalent by proving that the conditional statements $p1 \rightarrow p4$ , $p3 \rightarrow p1$ ,$p4 \rightarrow p2$ ,$p2 \rightarrow p5$, and $p5 \rightarrow p3$ are true.

Show that the propositions $p1,p2,p3,p4,$ and $p5$ can be shown to be equivalent by proving that the conditional statements $p1 \rightarrow p4$ , $p3 \rightarrow p1$ ,$p4...

Pooja Khatri

287 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

701

Kenneth Rosen Edition 7 Exercise 1.7 Question 36 (Page No. 91)
Show that the propositions $p1,p2,p3$, and $p4$can be shown to be equivalent by showing that $p1 \leftrightarrow p4,p2 \leftrightarrow p3$, and $p1 \leftrightarrow p3$.

Show that the propositions $p1,p2,p3$, and $p4$can be shown to be equivalent by showing that $p1 \leftrightarrow p4,p2 \leftrightarrow p3$, and $p1 \leftrightarrow p3$.

Pooja Khatri

230 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

702

Kenneth Rosen Edition 7 Exercise 1.7 Question 35 (Page No. 91)
Are these steps for finding the solutions of $\sqrt{x+3=3−x}$ correct? $\sqrt{x+3=3−x}$ is given; $x+3=x2−6x+9$, obtained by squaring both sides of(1); $0=x2−7x+6$, obtained by subtracting $x+3$ from both sides of(2); $0=(x−1)(x−6)$, ... -hand side of(3); $x=1$ or $x=6$,which follows from(4) because $ab=0$ implies that $a=0$ or $b=0$.

Are these steps for finding the solutions of $\sqrt{x+3=3−x}$ correct?$\sqrt{x+3=3−x}$ is given;$x+3=x2−6x+9$, obtained by squaring both sides of(1);$0=x2−7x+6$, ...

Pooja Khatri

262 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

703

Kenneth Rosen Edition 7 Exercise 1.7 Question 34 (Page No. 91)
Is this reasoning for finding the solutions of the equation $\sqrt{2x^2−1=x}$ correct? $\sqrt{2x^2−1=x}$ is given; $2x^2−1=x^2$, obtained by squaring both sides of (1); $x^2−1=0$, obtained by subtracting $x^2$ ... left-hand side of$x^2−1$; $x=1$ or $x=−1$,which follows because $ab=0$ implies that $a=0$ or $b=0$

Is this reasoning for finding the solutions of the equation $\sqrt{2x^2−1=x}$ correct?$\sqrt{2x^2−1=x}$ is given;$2x^2−1=x^2$, obtained by squaring both sides of (1...

Pooja Khatri

268 views

Pooja Khatri asked Apr 4, 2019

0 votes

1 answer

704

Kenneth Rosen Edition 7 Exercise 1.7 Question 33 (Page No. 91)
Show that these statements about the real number $x$ are equivalent: $x$ is irrational, $3x+2$ is irrational, $x/2$ is irrational.

Show that these statements about the real number $x$ are equivalent:$x$ is irrational,$3x+2$ is irrational,$x/2$ is irrational.

Pooja Khatri

598 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

705

Kenneth Rosen Edition 7 Exercise 1.7 Question 32 (Page No. 91)
Show that these statements about the real number $x$ are equivalent: $x$ is rational, $x/2$ is rational, $3x−1$ is rational.

Show that these statements about the real number $x$ are equivalent:$x$ is rational,$x/2$ is rational,$3x−1$ is rational.

Pooja Khatri

199 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

706

Kenneth Rosen Edition 7 Exercise 1.7 Question 31 (Page No. 91)
Show that these statements about the integer $x$ are equivalent: $3x+2$ is even, $x+5$ is odd, $x^2$ is even

Show that these statements about the integer $x$ are equivalent:$3x+2$ is even,$x+5$ is odd,$x^2$ is even

Pooja Khatri

257 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

707

Kenneth Rosen Edition 7 Exercise 1.7 Question 30 (Page No. 91)
Show that these three statements are equivalent, where $a$ and $b$ are real numbers: $a$ is less than $b$, the average of $a$ and $b$ is greater than $a$, and the average of $a$ and $b$ is less than $b$.

Show that these three statements are equivalent, where $a$ and $b$ are real numbers:$a$ is less than $b$,the average of $a$ and $b$ is greater than $a$, andthe average of...

Pooja Khatri

221 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

708

Kenneth Rosen Edition 7 Exercise 1.7 Question 29 (Page No. 91)
Prove or disprove that if $m$ and $n$ are integers such that $mn=1$, then either $m=1$ and $n=1$, or else $m=−1$ and $n=−1$.

Prove or disprove that if $m$ and $n$ are integers such that $mn=1$, then either $m=1$ and $n=1$, or else $m=−1$ and $n=−1$.

Pooja Khatri

201 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

709

Kenneth Rosen Edition 7 Exercise 1.7 Question 28 (Page No. 91)
Prove that $m^2 = n^2$ if and only if $m=n$ or m = -n.

Prove that $m^2 = n^2$ if and only if $m=n$ or m = -n.

Pooja Khatri

205 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

710

Kenneth Rosen Edition 7 Exercise 1.7 Question 27 (Page No. 91)
Prove that if $n$ is a positive integer, then $n$ is odd if and only if $5n+6$ is odd.

Prove that if $n$ is a positive integer, then $n$ is odd if and only if $5n+6$ is odd.

Pooja Khatri

178 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

711

Kenneth Rosen Edition 7 Exercise 1.7 Question 26 (Page No. 91)
Prove that if $n$ is a positive integer, then $n$ is even if and only if $7n+4$ is even.

Prove that if $n$ is a positive integer, then $n$ is even if and only if $7n+4$ is even.

Pooja Khatri

195 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

712

Kenneth Rosen Edition 7 Exercise 1.7 Question 25 (Page No. 91)
Use a proof by contradiction to show that there is no rational number $r$ for which $r^3+r+1=0$. [Hint:Assume that $r=a/b$ is a root, where $a$ and $b$ are integers and $a/b$ is in lowest terms. Obtain an equation involving integer $s$ by multiplying by $b^3$. Then look at whether $a$ and $b$ are each odd or even.]

Use a proof by contradiction to show that there is no rational number $r$ for which $r^3+r+1=0$. [Hint:Assume that $r=a/b$ is a root, where $a$ and $b$ are integers and $...

Pooja Khatri

235 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

713

Kenneth Rosen Edition 7 Exercise 1.7 Question 24 (Page No. 91)
Show that at least three of any $25$ days chosen must fall in the same month of the year.

Show that at least three of any $25$ days chosen must fall in the same month of the year.

Pooja Khatri

199 views

Pooja Khatri asked Apr 4, 2019

1 votes

0 answers

714

Kenneth Rosen Edition 7 Exercise 1.7 Question 23 (Page No. 91)
Show that at least ten of any $64$ days chosen must fall on the same day of the week.

Show that at least ten of any $64$ days chosen must fall on the same day of the week.

Pooja Khatri

211 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

715

Kenneth Rosen Edition 7 Exercise 1.7 Question 22 (Page No. 91)
Show that if you pick three socks from a drawer containing just blue socks and black socks, you must get either a pair of blue socks or a pair of black socks.

Show that if you pick three socks from a drawer containing just blue socks and black socks, you must get either a pair of blue socks or a pair of black socks.

Pooja Khatri

273 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

716

Kenneth Rosen Edition 7 Exercise 1.7 Question 21 (Page No. 91)
Let $P(n)$ be the proposition “If $a$ and $b$ are positive real numbers, then $(a+b)n≥a^n+b^n.$” Prove that $P(1)$ is true. What kind of proof did you use?

Let $P(n)$ be the proposition “If $a$ and $b$ are positive real numbers, then $(a+b)n≥a^n+b^n.$” Prove that $P(1)$ is true. What kind of proof did you use?

Pooja Khatri

197 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

717

Kenneth Rosen Edition 7 Exercise 1.7 Question 20 (Page No. 91)
Prove the position $P(1)$, where $P(n)$ is the proposition “If $n$ is a positive integer greater than $1$, then $n^2 > n.$” What kind of proof did you use?

Prove the position $P(1)$, where $P(n)$ is the proposition “If $n$ is a positive integer greater than $1$, then $n^2 n.$” What kind of proof did you use?

Pooja Khatri

231 views

Pooja Khatri asked Apr 4, 2019

1 votes

0 answers

718

Kenneth Rosen Edition 7 Exercise 1.7 Question 19 (Page No. 91)
Prove the position $P(0)$, where $P(n)$ is the proposition “If $n$ is a positive integer greater than $1$, then $n^2 > n.$” What kind of proof did you use?

Prove the position $P(0)$, where $P(n)$ is the proposition “If $n$ is a positive integer greater than $1$, then $n^2 n.$” What kind of proof did you use?

Pooja Khatri

259 views

Pooja Khatri asked Apr 4, 2019

0 votes

1 answer

719

Kenneth Rosen Edition 7 Exercise 1.7 Question 17 (Page No. 91)
Show that if $n$ is an integer and $n^3+5$ is odd, then $n$ is even using. a proof by contraposition. a proof by contradiction.

Show that if $n$ is an integer and $n^3+5$ is odd, then $n$ is even using.a proof by contraposition.a proof by contradiction.

Pooja Khatri

2.3k views

Pooja Khatri asked Apr 4, 2019

0 votes

1 answer

720

Kenneth Rosen Edition 7 Exercise 1.7 Question 16 (Page No. 91)
Prove that if $m$ and $n$ are integers and $mn$ is even, then $m$ is even or $n$ is even.

Prove that if $m$ and $n$ are integers and $mn$ is even, then $m$ is even or $n$ is even.

Pooja Khatri

284 views

Pooja Khatri asked Apr 4, 2019

tags	tag:apple
author	user:martin
title	title:apple
content	content:apple
exclude	-tag:apple
force match	+apple
views	views:100
score	score:10
answers	answers:2
is accepted	isaccepted:true
is closed	isclosed:true