Recent questions and answers in Mathematical Logic - GATE CSE Doubts

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pigeon hole principle
Each of 15 red balls and 15 green balls is marked with an integer between 1 and 100 inclusive; no integer appears on more than one ball. The value of a pair of balls is the sum of the numbers on the balls. Show there are at least two pairs, consisting of one red and one green ball, with the same value. Show that this is not true if there are 13 balls of each color.

asked 22 hours ago in Mathematical Logic by divi2719 (7 points) | 7 views

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PROBABILTY AND DISTRIBUTIONS
A elevator manufacturing company believes that 'X' is the amount of that can elevator withstand without any damage with is mean 100 and standard deviation 10. This elevator is used to lift the company staff persons with mean 5 and standard deviation 0.5. How many staff person would have to be in the elevator for the probability of No damage exceeds to 0.85.

asked Jun 30 in Mathematical Logic by Stanfordboi (6 points) | 8 views

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Combinatorics Simple doubt
What is the difference between flipping a pair of Distinct dices and flipping a pair of Identical Dices ??

asked Jun 18 in Mathematical Logic by BHASHKAR (47 points) | 10 views

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Kenneth Rosen 7th edition chapter 1.5 Exercise 12
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 students in your class. Use quantiﬁers ... to ask what will be the answer if statement means there are exactly two students who have not chatted with each other .

asked May 27 in Mathematical Logic by ayush.5 (13 points) | 8 views

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If A is a square matrix of order n, then number of elementary product of A.

asked May 14 in Mathematical Logic by iamHarin (8 points) | 11 views

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Tree self doubt
A certain tree of order n has only vertices of degree 1 and degree 3. How many degree-3 vertices does the tree have? (A). (B). (C). (D).

asked May 8 in Mathematical Logic by Abhipsa (27 points) | 15 views

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Havell Hakimi Algorithm - Graph Theory
The Havell Hakimi Algorithm Requires the sorting of the degree sequence and later marking and subtracting according to that order. Does this means that the vertex with the highest degree will always have an edge with the vertex with the second ... I have noticed that without sorting the algorithm doesn't give correct output so I think it is a necessary step)

asked May 5 in Mathematical Logic by nilotpola (7 points) | 32 views

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Discrete Mathematics - Set Theory
Can we say that total order relations are same as algebraic structure as in both the cases, we are enclosing the structures under some operation?

asked May 4 in Mathematical Logic by roh (17 points) | 3 views

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Cse zeal modules
How many subsets A of {1,2 3,....,10} have the property that no two elements of A sum to 11

asked Apr 25 in Mathematical Logic by Syywyeye (9 points) | 19 views

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This question is from gate cse zeal acadmey module

asked Apr 25 in Mathematical Logic by Syywyeye (9 points) | 15 views

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GATE2018-28 Video Solution
Consider the first-order logic sentence $\varphi \equiv \exists \: s \: \exists \: t \: \exists \: u \: \forall \: v \: \forall \: w \forall \: x \: \forall \: y \: \psi(s, t, u, v, w, x, y)$ ... or equal to $3$ There exists no model of $\varphi$ with universe size of greater than $7$ Every model of $\varphi$ has a universe of size equal to $7$

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 8 views

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GATE2016-2-01 Video Solution
Consider the following expressions: $false$ $Q$ $true$ $P\vee Q$ $\neg Q\vee P$ The number of expressions given above that are logically implied by $P \wedge (P \Rightarrow Q)$ is ___________.

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 3 views

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GATE2015-2-55 Video Solution
Which one of the following well-formed formulae is a tautology? $\forall x \, \exists y \, R(x,y) \, \leftrightarrow \, \exists y \, \forall x \, R(x, y)$ ... $\forall x \, \forall y \, P(x,y) \, \rightarrow \, \forall x \, \forall y \, P(y, x)$

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2019-35 Video Solution
Consider the first order predicate formula $\varphi$: $\forall x [ ( \forall z \: z \mid x \Rightarrow (( z=x) \vee (z=1))) \rightarrow \exists w ( w > x) \wedge (\forall z \: z \mid w \Rightarrow ((w=z) \vee (z=1)))]$ ... of all positive integers $S3:$ Set of all integers Which of the above sets satisfy $\varphi$? S1 and S2 S1 and S3 S2 and S3 S1, S2 and S3

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2016-2-27 Video Solution
Which one of the following well-formed formulae in predicate calculus is NOT valid ? $(\forall _{x} p(x) \implies \forall _{x} q(x)) \implies (\exists _{x} \neg p(x) \vee \forall _{x} q(x))$ ... $\forall x (p(x) \vee q(x)) \implies (\forall x p(x) \vee \forall x q(x))$

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2017-1-02 Video Solution
Consider the first-order logic sentence $F:\forall x(\exists yR(x,y))$. Assuming non-empty logical domains, which of the sentences below are implied by $F$? $\exists y(\exists xR(x,y))$ $\exists y(\forall xR(x,y))$ $\forall y(\exists xR(x,y))$ $¬\exists x(\forall y¬R(x,y))$ IV only I and IV only II only II and III only

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 3 views

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GATE2003-32 Video Solution
Which of the following is a valid first order formula? (Here \(\alpha\) and \(\beta\) are first order formulae with $x$ as their only free variable) $((∀x)[α] ⇒ (∀x)[β]) ⇒ (∀x)[α ⇒ β]$ $(∀x)[α] ⇒ (∃x)[α ∧ β]$ $((∀x)[α ∨ β] ⇒ (∃x)[α]) ⇒ (∀x)[α]$ $(∀x)[α ⇒ β] ⇒ (((∀x)[α]) ⇒ (∀x)[β])$

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2015-3-24 Video Solution
In a room there are only two types of people, namely $\text{Type 1}$ and $\text{Type 2}$. $\text{Type 1}$ people always tell the truth and $\text{Type 2}$ people always lie. You give a fair coin to a person in that room, without knowing which type he ... person is of $\text{Type 2}$, then the result is tail If the person is of $\text{Type 1}$, then the result is tail

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE1992-92,xv Video Solution
Which of the following predicate calculus statements is/are valid? $(\forall (x)) P(x) \vee (\forall(x))Q(x) \implies (\forall (x)) (P(x) \vee Q(x))$ $(\exists (x)) P(x) \wedge (\exists (x))Q(x) \implies (\exists (x)) (P(x) \wedge Q(x))$ ... $(\exists (x)) (P(x) \vee Q(x)) \implies \sim (\forall (x)) P(x) \vee (\exists (x)) Q(x)$

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 3 views

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GATE2004-23, ISRO2007-32 Video Solution
Identify the correct translation into logical notation of the following assertion. Some boys in the class are taller than all the girls Note: $\text{taller} (x, y)$ is true if $x$ is taller than $y$ ... $(\exists x) (\text{boy}(x) \land (\forall y) (\text{girl}(y) \rightarrow \text{taller}(x, y)))$

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 8 views

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GATE2016-1-1 Video Solution
Let $p, q, r, s$ represents the following propositions. $p:x\in\left\{8, 9, 10, 11, 12\right\}$ $q:$ $x$ is a composite number. $r:$ $x$ is a perfect square. $s:$ $x$ is a prime number. The integer $x\geq2$ which satisfies $\neg\left(\left(p\Rightarrow q\right) \wedge \left(\neg r \vee \neg s\right)\right)$ is ____________.

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2010-30 Video Solution
Suppose the predicate $F(x, y, t)$ is used to represent the statement that person $x$ can fool person $y$ at time $t$. Which one of the statements below expresses best the meaning of the formula, $\qquad∀x∃y∃t(¬F(x,y,t))$ Everyone can fool ... time No one can fool everyone all the time Everyone cannot fool some person all the time No one can fool some person at some time

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 3 views

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GATE2002-1.8 Video Solution
"If $X$ then $Y$ unless $Z$" is represented by which of the following formulas in prepositional logic? ("$\neg$" is negation, "$\land$" is conjunction, and "$\rightarrow$" is implication) $(X\land \neg Z) \rightarrow Y$ $(X \land Y) \rightarrow \neg Z$ $X \rightarrow(Y\land \neg Z)$ $(X \rightarrow Y)\land \neg Z$

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 1 view

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GATE2003-33 Video Solution
Consider the following formula and its two interpretations \(I_1\) and \(I_2\). \(\alpha: (\forall x)\left[P_x \Leftrightarrow (\forall y)\left[Q_{xy} \Leftrightarrow \neg Q_{yy} \right]\right] \Rightarrow (\forall x)\left[\neg P_x\right]\) \(I_1\) : Domain: ... (I_1\) does not Neither \(I_1\) nor \(I_2\) satisfies \(\alpha\) Both \(I_1\) and \(I_2\) satisfies \(\alpha\)

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2005-IT-36 Video Solution
Let $P(x)$ and $Q(x)$ ...

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2008-IT-21 Video Solution
Which of the following first order formulae is logically valid? Here $\alpha(x)$ is a first order formula with $x$ as a free variable, and $\beta$ ... $[(\forall x, \alpha(x)) \rightarrow \beta] \rightarrow [\forall x, \alpha(x) \rightarrow \beta]$

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 3 views

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GATE2006-IT-21 Video Solution
Consider the following first order logic formula in which $R$ is a binary relation symbol. $∀x∀y (R(x, y) \implies R(y, x))$ The formula is satisfiable and valid satisfiable and so is its negation unsatisfiable but its negation is valid satisfiable but its negation is unsatisfiable

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2008-30 Video Solution
Let $\text{fsa}$ and $\text{pda}$ be two predicates such that $\text{fsa}(x)$ means $x$ is a finite state automaton and $\text{pda}(y)$ means that $y$ is a pushdown automaton. Let $\text{equivalent}$ ...

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2017-2-11 Video Solution
Let $p, q, r$ ... $(\neg p \wedge r) \vee (r \rightarrow (p \wedge q))$

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2011-30 Video Solution
Which one of the following options is CORRECT given three positive integers $x, y$ and $z$ ... always true irrespective of the value of $x$ $P(x)$ being true means that $x$ has exactly two factors other than $1$ and $x$

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 1 view

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GATE2013-47 Video Solution
Which one of the following is NOT logically equivalent to $¬∃x(∀ y (α)∧∀z(β ))$ ? $∀ x(∃ z(¬β )→∀ y(α))$ $∀x(∀ z(β )→∃ y(¬α))$ $∀x(∀ y(α)→∃z(¬β ))$ $∀x(∃ y(¬α)→∃z(¬β ))$

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2014-1-53 Video Solution
Which one of the following propositional logic formulas is TRUE when exactly two of $p,q$ and $r$ are TRUE? $(( p \leftrightarrow q) \wedge r) \vee (p \wedge q \wedge \sim r)$ $( \sim (p \leftrightarrow q) \wedge r)\vee (p \wedge q \wedge \sim r)$ ... $(\sim (p \leftrightarrow q) \wedge r) \wedge (p \wedge q \wedge \sim r) $

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 1 view

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GATE1998-1.5 Video Solution
What is the converse of the following assertion? I stay only if you go I stay if you go If I stay then you go If you do not go then I do not stay If I do not stay then you go

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 1 view

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GATE2013-27 Video Solution
What is the logical translation of the following statement? "None of my friends are perfect." $∃x(F (x)∧ ¬P(x))$ $∃ x(¬ F (x)∧ P(x))$ $ ∃x(¬F (x)∧¬P(x))$ $ ¬∃ x(F (x)∧ P(x))$

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2003-72 Video Solution
The following resolution rule is used in logic programming. Derive clause $(P \vee Q)$ from clauses $(P\vee R),(Q \vee ¬R)$ Which of the following statements related to this rule is FALSE? $((P ∨ R)∧(Q ∨ ¬R))⇒(P ∨ Q)$ ... only if $(P ∨ R)∧(Q ∨ ¬R)$ is satisfiable $(P ∨ Q)⇒ \text{FALSE}$ if and only if both $P$ and $Q$ are unsatisfiable

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 1 view

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GATE2017-1-29 Video Solution
Let $p$, $q$ and $r$ be propositions and the expression $\left ( p\rightarrow q \right )\rightarrow r$ be a contradiction. Then, the expression $\left ( r\rightarrow p \right )\rightarrow q$ is a tautology a contradiction always TRUE when $p$ is FALSE always TRUE when $q$ is TRUE

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 1 view

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GATE2005-41 Video Solution
What is the first order predicate calculus statement equivalent to the following? "Every teacher is liked by some student" $∀(x)\left[\text{teacher}\left(x\right) → ∃(y) \left[\text{student}\left(y\right) → \text{likes}\left(y,x\right)\right]\right]$ ...

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 3 views

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GATE2007-22 Video Solution
$\def\graph{\text{ Graph}} \def\connected{\text{ Connected}}$ Let $\graph(x)$ be a predicate which denotes that $x$ is a graph. Let $\connected(x)$ be a predicate which denotes that $x$ is connected. Which of the following first order logic sentences DOES NOT ... $\forall x \, \Bigl ( \graph(x) \implies \lnot \connected(x) \Bigr )$

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2017-1-01 Video Solution
The statement $\left ( ¬p \right ) \Rightarrow \left ( ¬q \right )$ is logically equivalent to which of the statements below? $p \Rightarrow q$ $q \Rightarrow p$ $\left ( ¬q \right ) \vee p$ $\left ( ¬p \right ) \vee q$ I only I and IV only II only II and III only

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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GATE2014-3-1 Video Solution
Consider the following statements: P: Good mobile phones are not cheap Q: Cheap mobile phones are not good L: P implies Q M: Q implies P N: P is equivalent to Q Which one of the following about L, M, and N is CORRECT? Only L is TRUE. Only M is TRUE. Only N is TRUE. L, M and N are TRUE.

asked Apr 19 in Mathematical Logic by admin (3.6k points) | 2 views

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