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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 bucketshaped 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 nonoverlapping trapeziums.
asked
May 8
in
Study Resources
by
sumitsehgal
(
8
points)

8
views
mathematicallogic
0
votes
0
answers
GATE201828 Video Solution
Consider the firstorder 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
gate2018
mathematicallogic
normal
firstorderlogic
videosolution
0
votes
0
answers
GATE2016201 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
gate20162
mathematicallogic
normal
numericalanswers
propositionallogic
videosolution
0
votes
0
answers
GATE2015255 Video Solution
Which one of the following wellformed 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
gate20152
mathematicallogic
normal
firstorderlogic
videosolution
0
votes
0
answers
GATE201935 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
gate2019
engineeringmathematics
discretemathematics
mathematicallogic
firstorderlogic
videosolution
0
votes
0
answers
GATE2016227 Video Solution
Which one of the following wellformed 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
gate20162
mathematicallogic
firstorderlogic
normal
videosolution
0
votes
0
answers
GATE2017102 Video Solution
Consider the firstorder logic sentence $F:\forall x(\exists yR(x,y))$. Assuming nonempty 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
gate20171
mathematicallogic
firstorderlogic
videosolution
0
votes
0
answers
GATE200332 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
gate2003
mathematicallogic
firstorderlogic
normal
videosolution
0
votes
0
answers
GATE2015324 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
gate20153
mathematicallogic
difficult
logicalreasoning
videosolution
0
votes
0
answers
GATE199292,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
gate1992
mathematicallogic
normal
firstorderlogic
videosolution
0
votes
0
answers
GATE200423, ISRO200732 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
gate2004
mathematicallogic
easy
isro2007
firstorderlogic
videosolution
0
votes
0
answers
GATE201611 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
gate20161
mathematicallogic
normal
numericalanswers
propositionallogic
videosolution
0
votes
0
answers
GATE201030 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
gate2010
mathematicallogic
easy
firstorderlogic
videosolution
0
votes
0
answers
GATE20021.8 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
gate2002
mathematicallogic
normal
propositionallogic
videosolution
0
votes
0
answers
GATE200333 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
gate2003
mathematicallogic
difficult
firstorderlogic
videosolution
0
votes
0
answers
GATE2005IT36 Video Solution
Let $P(x)$ and $Q(x)$ ...
asked
Apr 19
in
Mathematical Logic
by
admin
(
3.6k
points)

2
views
gate2005it
mathematicallogic
firstorderlogic
normal
videosolution
0
votes
0
answers
GATE2006IT21 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
gate2006it
mathematicallogic
normal
firstorderlogic
videosolution
0
votes
0
answers
GATE200830 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
gate2008
easy
mathematicallogic
firstorderlogic
videosolution
0
votes
0
answers
GATE2017211 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
gate20172
mathematicallogic
propositionallogic
videosolution
0
votes
0
answers
GATE201130 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
gate2011
mathematicallogic
normal
firstorderlogic
videosolution
0
votes
0
answers
GATE201347 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
mathematicallogic
normal
markstoall
gate2013
firstorderlogic
videosolution
0
votes
0
answers
GATE2014153 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
gate20141
mathematicallogic
normal
propositionallogic
videosolution
0
votes
0
answers
GATE19981.5 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
gate1998
mathematicallogic
easy
propositionallogic
videosolution
0
votes
0
answers
GATE201327 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
gate2013
mathematicallogic
easy
firstorderlogic
videosolution
0
votes
0
answers
GATE200372 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
gate2003
mathematicallogic
normal
propositionallogic
videosolution
0
votes
0
answers
GATE2017129 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
gate20171
mathematicallogic
propositionallogic
videosolution
0
votes
0
answers
GATE200541 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
gate2005
mathematicallogic
easy
firstorderlogic
videosolution
0
votes
0
answers
GATE200722 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
gate2007
mathematicallogic
easy
firstorderlogic
videosolution
0
votes
0
answers
GATE2017101 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
gate20171
mathematicallogic
propositionallogic
easy
videosolution
0
votes
0
answers
GATE201431 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
gate20143
mathematicallogic
easy
propositionallogic
videosolution
0
votes
0
answers
GATE2007IT21 Video Solution
Which one of these firstorder 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
gate2007it
mathematicallogic
normal
firstorderlogic
videosolution
0
votes
0
answers
GATE2004IT31 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
gate2004it
mathematicallogic
normal
propositionallogic
videosolution
0
votes
0
answers
GATE2014253 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
gate20142
mathematicallogic
propositionallogic
normal
videosolution
0
votes
0
answers
GATE20002.7 Video Solution
Let $a, b, c, d$ be propositions. Assume that the equivalence $a ⇔ ( b \vee \neg b)$ and $b ⇔c$ hold. Then the truthvalue of the formula $(a ∧ b) → (a ∧ c) ∨ d$ is always True False Same as the truthvalue of $b$ Same as the truthvalue of $d$
asked
Apr 19
in
Mathematical Logic
by
admin
(
3.6k
points)

1
view
gate2000
mathematicallogic
normal
propositionallogic
videosolution
0
votes
0
answers
GATE20011.3 Video Solution
Consider two wellformed 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
gate2001
mathematicallogic
easy
propositionallogic
videosolution
0
votes
0
answers
GATE200626 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
gate2006
mathematicallogic
normal
firstorderlogic
videosolution
0
votes
0
answers
GATE201523 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
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admin
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3.6k
points)

1
view
gate20152
mathematicallogic
normal
logicalreasoning
videosolution
0
votes
0
answers
GATE2015114 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
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admin
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3.6k
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1
view
gate20151
mathematicallogic
easy
propositionallogic
videosolution
0
votes
0
answers
GATE20121 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
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Apr 19
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Mathematical Logic
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admin
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3.6k
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8
views
gate2012
mathematicallogic
easy
logicalreasoning
videosolution
0
votes
0
answers
GATE19952.19 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
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Mathematical Logic
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admin
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3.6k
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1
view
gate1995
mathematicallogic
normal
propositionallogic
videosolution
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