Recent questions tagged mathematical-logic

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Mathematical Logic Ques:Self doubt
“Not every satisfiable logic is valid” Representation of it will be $1)\sim \left ( \forall S(x)\rightarrow V(x) \right )$ or $2)\sim \left ( \forall S(x)\vee V(x) \right )$ Among $1)$ and $2)$, which one is correct? and why?

asked Jun 4 in Mathematical Logic by srestha Veteran (112k points) | 97 views

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Doubt on GATE Question
Read the statements: All women are entrepreneurs. Some women are doctors. Which of the following conclusions can be logically inferred from the above statements? All women are doctors All doctors are entrepreneurs All entrepreneurs are women Some entrepreneurs are doctors ... Is it because , if we make set of doctor as 0, then All doctors are entrepreneurs is meaningless.

asked Jun 1 in Mathematical Logic by srestha Veteran (112k points) | 39 views

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Mathematical Logic: Doubt on meaning of statement
The notation $\exists ! x P(x)$ denotes the proposition there exists a unique $x$ such that $P(x)$ ... What will be answer here?? Is the assumption only for left hand side and not right hand side??

asked May 31 in Mathematical Logic by srestha Veteran (112k points) | 71 views

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Proposition Logic Question
Are these propositions? 1.This sentence is true 2.This sentence is false Aren’t these liar paradox?

asked May 30 in Mathematical Logic by Reshu $ingh (253 points) | 104 views

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Self Doubt:Mathematical Logic
Represent these two statement in first order logic: $A)$ Only Alligators eat humans $B)$ Every Alligator eats humans Is Every represents $\equiv \exists$ and Only represents $\equiv \forall$ ?? Can we differentiate it with verb ‘eat’ and ‘eats’??

asked May 18 in Mathematical Logic by srestha Veteran (112k points) | 40 views

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Discrete mathematics and its application 7th ed - Kenneth H. Rosen
Do i have to study the whole chapter Logics and Proofs in Discrete mathematics and its applications by Kenneth H. Rosen if not upto which portion should i study.

asked May 1 in Mathematical Logic by souren (37 points) | 45 views

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Made Easy Test Series:Discrete Math-Mathematical Logic
Consider the following first order logic statement $I)\forall x\forall yP\left ( x,y \right )$ $II)\forall x\exists yP\left ( x,y \right )$ $III)\exists x\exists yP\left ( x,y \right )$ $III)\exists x\forall yP\left ( x,y \right )$ Which one ... true , then $III),IV)$ is true $B)$ If $IV)$ is true , then $II),III)$ is true $C)$ None of these

asked Apr 27 in Mathematical Logic by srestha Veteran (112k points) | 47 views

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Kenneth Rosen Edition 7th 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$} $\phi$ $\subset$ {$0$} {$0$} $\epsilon$ {$0$} {$0$} $\subset$ {$0$} {$\phi$} $\subseteq$ {$\phi$}

asked Apr 5 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 38 views

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Kenneth Rosen Edition 7th 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.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 25 views

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Kenneth Rosen Edition 7th 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.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 11 views

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Kenneth Rosen Edition 7th 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?

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 13 views

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Kenneth Rosen Edition 7th 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

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 13 views

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Kenneth Rosen Edition 7th 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.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 20 views

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Kenneth Rosen Edition 7th 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 $p1 \leftrightarrow p3$.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 12 views

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Kenneth Rosen Edition 7th 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 $b=0$.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 22 views

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Kenneth Rosen Edition 7th 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 $x^2$from both sides of (2); ... left-hand side of$x^2−1$; $x=1$ or $x=−1$,which follows because $ab=0$ implies that $a=0$ or $b=0$

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 11 views

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Kenneth Rosen Edition 7th 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.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 10 views

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Kenneth Rosen Edition 7th 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.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 9 views

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Kenneth Rosen Edition 7th 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

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 9 views

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Kenneth Rosen Edition 7th 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 $b$.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 9 views

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Kenneth Rosen Edition 7th 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 $n=−1$.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 7 views

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

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 8 views

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23

Kenneth Rosen Edition 7th 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.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 5 views

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Kenneth Rosen Edition 7th 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.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 8 views

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Kenneth Rosen Edition 7th 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.]

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 12 views

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Kenneth Rosen Edition 7th 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.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 8 views

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27

Kenneth Rosen Edition 7th 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.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 11 views

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28

Kenneth Rosen Edition 7th 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.

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 11 views

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Kenneth Rosen Edition 7th 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?

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 7 views

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Kenneth Rosen Edition 7th 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?

asked Apr 4 in Mathematical Logic by Pooja Khatri Boss (10.8k points) | 8 views

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