Recent questions tagged mathematical-logic - GATE CSE Doubts

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Trapeziums : Mathematics
An equilateral triangle, with side of length 3n for some natural number n, is made of smaller equilateral triangles. See the figure below for the case n=2. A bucket-shaped trapezium shown in the right of the below figure is made from three equilateral triangles. Prove that it is possible to cover the remaining triangles with non-overlapping trapeziums.

asked May 8 in Study Resources by sumitsehgal (8 points) | 8 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) | 2 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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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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GATE2007-IT-21 Video Solution
Which one of these first-order logic formulae is valid? $\forall x\left(P\left(x\right) \implies Q\left(x\right)\right) \implies \left(∀xP\left(x\right)\implies \forall xQ\left(x\right)\right)$ ... $\forall x \exists y P\left(x, y\right)\implies \exists y \forall x P\left(x, y\right)$

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

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GATE2004-IT-31 Video Solution
Let $p, q, r$ and $s$ be four primitive statements. Consider the following arguments: $P: [(¬p\vee q) ∧ (r → s) ∧ (p \vee r)] → (¬s → q)$ $Q: [(¬p ∧q) ∧ [q → (p → r)]] → ¬r$ $R: [[(q ∧ r) → p] ∧ (¬q \vee p)] → r$ $S: [p ∧ (p → r) ∧ (q \vee ¬ r)] → q$ Which of the above arguments are valid? $P$ and $Q$ only $P$ and $R$ only $P$ and $S$ only $P, Q, R$ and $S$

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

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GATE2014-2-53 Video Solution
Which one of the following Boolean expressions is NOT a tautology? $((\,a\,\to\,b\,)\,\wedge\,(\,b\,\to\,c))\,\to\,(\,a\,\to\,c)$ $(\,a\,\to\,c\,)\,\to\,(\,\sim b\,\to\,(a\,\wedge\,c))$ $(\,a\,\wedge\,b\,\wedge\,c)\,\to\,(\,c\vee\,a)$ $a\,\to\,(b\,\to\,a)$

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

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GATE2000-2.7 Video Solution
Let $a, b, c, d$ be propositions. Assume that the equivalence $a ⇔ ( b \vee \neg b)$ and $b ⇔c$ hold. Then the truth-value of the formula $(a ∧ b) → (a ∧ c) ∨ d$ is always True False Same as the truth-value of $b$ Same as the truth-value of $d$

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

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GATE2001-1.3 Video Solution
Consider two well-formed formulas in propositional logic $F_1: P \Rightarrow \neg P$ $F_2: (P \Rightarrow \neg P) \lor ( \neg P \Rightarrow P)$ Which one of the following statements is correct? $F_1$ is satisfiable, $F_2$ is valid $F_1$ unsatisfiable, $F_2$ is satisfiable $F_1$ is unsatisfiable, $F_2$ is valid $F_1$ and $F_2$ are both satisfiable

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

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GATE2006-26 Video Solution
Which one of the first order predicate calculus statements given below correctly expresses the following English statement? Tigers and lions attack if they are hungry or threatened. ...

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

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GATE2015-2-3 Video Solution
Consider the following two statements. $S_1$: If a candidate is known to be corrupt, then he will not be elected $S_2$: If a candidate is kind, he will be elected Which one of the following statements follows from $S_1$ and $S_2$ as per sound inference rules ... If a person is kind, he is not known to be corrupt If a person is not kind, he is not known to be corrupt

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

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GATE2015-1-1‏4 Video Solution
Which one of the following is NOT equivalent to $p ↔ q$? $(\neg p ∨ q) ∧ (p ∨ \neg q)$ $(\neg p ∨ q) ∧ (q → p)$ $(\neg p ∧ q) ∨ ( p ∧ \neg q)$ $(\neg p ∧ \neg q) ∨ (p ∧ q)$

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

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GATE2012-1 Video Solution
Consider the following logical inferences. $I_{1}$: If it rains then the cricket match will not be played. The cricket match was played. Inference: There was no rain. $I_{2}$: If it rains then the cricket match will not be played. It did not rain. Inference: The ... $I_{2}$ is a correct inference Both $I_{1}$ and $I_{2}$ are not correct inferences

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

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GATE1995-2.19 Video Solution
If the proposition $\lnot p \to q$ is true, then the truth value of the proposition $\lnot p \lor \left ( p \to q \right )$, where $\lnot$ is negation, $\lor$ is inclusive OR and $\to$ is implication, is True Multiple Values False Cannot be determined

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

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