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

0 votes

1 answer

961

Kenneth Rosen Edition 7 Exercise 2.1 Question 10 (Page No. 125)
Determine whether each of these statements is true or false. $\phi$ $ \epsilon$ {$\phi$} $\phi$ $\epsilon$ {$\phi,$ { $\phi$}} {$\phi$} $ \epsilon$ {$ \phi$} {$\phi$} $\epsilon $ {{$\phi$}} {$\phi$} $\subset$ {$0$} {$0$} $\subset$ {$\phi$ , { $\phi$ }} {$\phi$} $\subset$ {{$\phi$ }, { $\phi$}}

Determine whether each of these statements is true or false.$\phi$ $ \epsilon$ {$\phi$}$\phi$ $\epsilon$ {$\phi,$ { $\phi$}}{$\phi$} $ \epsilon$ {$ \phi$}{$\phi$} $\epsil...

Pooja Khatri

448 views

Pooja Khatri asked Apr 5, 2019

4 votes

2 answers

962

Kenneth Rosen Edition 7 Exercise 2.1 Question 9 (Page No. 125)
Determine whether each of these statements is true or false. $0$ $ \epsilon$ $\phi$ $\phi$ $\epsilon$ {$0$} {$0$} $ \subset$ {$ \phi$} $\phi$ $\subset$ {$0$} {$0$} $\epsilon$ {$0$} {$0$} $\subset$ {$0$} {$\phi$} $\subseteq$ {$\phi$}

Determine whether each of these statements is true or false.$0$ $ \epsilon$ $\phi$$\phi$ $\epsilon$ {$0$}{$0$} $ \subset$ {$ \phi$}$\phi$ $\subset$ {$0$}{$0$} $\epsilon$ ...

Pooja Khatri

488 views

Pooja Khatri asked Apr 5, 2019

0 votes

1 answer

963

Kenneth Rosen Edition 7 Exercise 2.1 Question 7 (Page No. 125)
For each of the following sets, determine whether 2 is an element of that set. $\{ x \in R \mid x \text{ is an integer greater than} \}$ $\{x \in R \mid x \text{ is the square of an integer}\}$ $\{ 2 ,\{ 2 \}\}$ $\{\{ 2\},\{\{ 2 \}\}\}$ $\{\{2 \},\{2,\{2 \}\}\}$ $\{\{\{2\}\}\}$

For each of the following sets, determine whether 2 is an element of that set.$\{ x \in R \mid x \text{ is an integer greater than} \}$$\{x \in R \mid x \text{ is the sq...

Pooja Khatri

297 views

Pooja Khatri asked Apr 5, 2019

0 votes

1 answer

964

Kenneth Rosen Edition 7 Exercise 2.1 Question 6 (Page No. 125)
Suppose that $A=$ { $2,4,6$ }, $B=$ { $2,6$ }, $C=$ { $4,6$ }, and $D=$ { $4,6,8$ }. Determine which of these sets are subsets of which other of these sets.

Suppose that $A=$ { $2,4,6$ }, $B=$ { $2,6$ }, $C=$ { $4,6$ }, and $D=$ { $4,6,8$ }. Determine which of these sets are subsets of which other of these sets.

Pooja Khatri

380 views

Pooja Khatri asked Apr 5, 2019

0 votes

0 answers

965

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

324 views

Pooja Khatri asked Apr 5, 2019

0 votes

0 answers

966

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

341 views

Pooja Khatri asked Apr 5, 2019

0 votes

0 answers

967

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

349 views

Pooja Khatri asked Apr 5, 2019

1 votes

0 answers

968

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

206 views

Pooja Khatri asked Apr 5, 2019

0 votes

0 answers

969

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

236 views

Pooja Khatri asked Apr 5, 2019

0 votes

0 answers

970

DDA Direct Recruitment 2019 - Asst. System Director
Which of the following gives the predicate logic representation of the sentence Ram was a man ? Ram $\rightarrow$ man $\forall x:$ Ram (x) $\rightarrow$ man (x) Man (Ram) Man $\rightarrow$ Ram I have marked option ... I want to raise an objection to this answer. Kindly provide a reliable source which confirms the correct answer for this question.

Which of the following gives the predicate logic representation of the sentence “Ram was a man”?Ram $\rightarrow$ man$\forall x:$ Ram (x) $\rightarrow$ man (x)Man (Ra...

zeeshanmohnavi

639 views

zeeshanmohnavi asked Apr 4, 2019

0 votes

0 answers

971

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

296 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

972

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

246 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

973

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

974

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

333 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

975

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

289 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

976

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

977

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

265 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

978

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

269 views

Pooja Khatri asked Apr 4, 2019

0 votes

1 answer

979

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

606 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

980

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

201 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

981

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

982

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

224 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

983

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

204 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

984

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

985

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

180 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

986

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

197 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

987

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

236 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

988

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

200 views

Pooja Khatri asked Apr 4, 2019

1 votes

0 answers

989

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

214 views

Pooja Khatri asked Apr 4, 2019

0 votes

0 answers

990

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

274 views

Pooja Khatri asked Apr 4, 2019

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