# Recent questions and answers in Mathematical Logic 1
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 is from and tell ... tail If the person is of $\text{Type 2}$, then the result is tail If the person is of $\text{Type 1}$, then the result is tail
1 vote
2
CONVERT IN TO LOGIC No one who loves some one is not loved by anyone lets S(x):x is somebody L(x,y):x loves y
3
Can the answer to this be "∀x ∃y (teacher (x) ∧ student (y) ∧ likes (y,x))" ?
4
$P$ and $Q$ are two propositions. Which of the following logical expressions are equivalent? $P ∨ \neg Q$ $\neg(\neg P ∧ Q)$ $(P ∧ Q) ∨ (P ∧ \neg Q) ∨ (\neg P ∧ \neg Q)$ $(P ∧ Q) ∨ (P ∧ \neg Q) ∨ (\neg P ∧ Q)$ Only I and II Only I, II and III Only I, II and IV All of I, II, III and IV
5
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 $\text{equivalent}$ be another predicate such that $\text{equivalent} (a,b)$ ...
1 vote
6
if $\lambda$3 - 6$\lambda$2 -$\lambda$ +22=0 is a characteristic of 3 X 3 diagonal matrix , then trace of matrix A is
7
Every satisfiable propositional formula is not tautology. True/False
8
1) IS P → Q ≡ Q → P Satisfiable Or NOT?
9
Statements: Some boxes are triangles. All Spheres are triangles. All circles are boxes. All triangles are quadrilaterals. Conclusions: Some quadrilaterals are boxes. Some quadrilaterals are triangles. Some triangles are spheres. No circle is quadrilaterals. Options. 1 only 1st follow 2 only 1 ,2 ,3 follows 3 only 3rd follows 4 all follows
10
In propositional logic, which of the following is equivalent to $p \rightarrow q$? $\sim p\rightarrow q$ $\sim p \vee q$ $\sim p \vee \sim q$ $p\rightarrow \sim q$
11
What is logically equivalent to "If Kareena and Parineeti go to the shopping mall then it is raining": If Kareena and Parineeti do not go to the shopping mall then it is not raining. If Kareena and Parineeti do not go to the shopping mall then it is raining. If it ... go to the shopping mall. If it is not raining then Kareena and Parineeti do not go to the shopping mall. None of the above.
12
Consider the following well-formed formulae: $\neg \forall x(P(x))$ $\neg \exists x(P(x))$ $\neg \exists x(\neg P(x))$ $\exists x(\neg P(x))$ Which of the above are equivalent? I and III I and IV II and III II and IV
1 vote
13
In a lottery, 10 tickets are drawn at random out of 50 tickets numbered from 1 to 50. What is the expected value of the sum of numbers on the drawn tickets?
14
In a non-homogeneous equation Ax = b, x has a unique solution when $A^{-1}$ exists i.e x = $A^{-1}$b but when det(A) = 0 then we have infinite solution or many solution. please give a mathematical explanation of how the 2nd statement occurs?
15
Which one of the following propositional logic formulas is TRUE when exactly two of $p,q$ and $r$ are TRUE? $(( p \leftrightarrow q) \wedge r) \vee (p \wedge q \wedge \sim r)$ $( \sim (p \leftrightarrow q) \wedge r)\vee (p \wedge q \wedge \sim r)$ $( (p \to q) \wedge r) \vee (p \wedge q \wedge \sim r)$ $(\sim (p \leftrightarrow q) \wedge r) \wedge (p \wedge q \wedge \sim r)$
16
1 vote
17
18
Let the number of non-isomorphic groups of order 10 be X and number of non-isomorphic groups of order 24 be Y then the value of X and Y a) 3,2 b)2,7 c)1,7 d)4,5
19
Consider the following first order logic statement $I)\forall x\forall yP\left ( x,y \right )$ $II)\forall x\exists yP\left ( x,y \right )$ $III)\exists x\exists yP\left ( x,y \right )$ $III)\exists x\forall yP\left ( x,y \right )$ ... $II)$ is true , then $III),IV)$ is true $B)$ If $IV)$ is true , then $II),III)$ is true $C)$ None of these
20
The formula for the number of positive integers m which are less than p^k and relatively prime to p^k, where p is a prime number and k is a positive integer is__________- A)p^k(p-1) B)(p^(k-2))(p-1) C)p^k(p-2) D)(p^(k-1))(p-1)
21
Three candidates, Amar, Birendra and Chanchal stand for the local election. Opinion polls are conducted and show that fraction $a$ of the voters prefer Amar to Birendra, fraction $b$ prefer Birendra to Chanchal and fraction $c$ ... $(a, b, c) = (0.49, 0.49, 0.49);$ None of the above.
22
Which one is the correct translation of the following statement into mathematical logic? “None of my friends are perfect.” $\neg\:\exists\:x(p(x)\land q(x))$ $\exists\:x(\neg\:p(x)\land q(x))$ $\exists\:x(\neg\:p(x)\land\neg\:q(x))$ $\exists\:x(p(x)\land\neg\:q(x))$
23
Some cat are intelligent express into first order logic if domain are animals
24
Consider the random variable X such that it takes values +1,-1 and +2 with probability 0.1 each .Calculate values of the commulative frequencydistribution function F(x) at x=-1 and x=1 and x=2 are ?
25
In the following, $A$ stands for a set of apples, and $S(x, y)$ stands for "$x$ is sweeter than $y$. Let $\Psi \equiv \exists x : x \in A$ $\Phi \equiv \forall x \in A : \exists y \in A : S(x, y).$ ...
26
Let X and Y be two exponentially distributed and independent random variables with mean α and β, respectively. If Z = max (X, Y), then the mean of Z is…. please explain in detail… https://gateoverflow.in/3676/gate2004-it-33 for min(X, Y) solution is already given as question asked in gate 2004. what about max(X, Y).
27
Consider the first order predicate formula $\varphi$: $\forall x [ ( \forall z \: z \mid x \Rightarrow (( z=x) \vee (z=1))) \rightarrow \exists w ( w > x) \wedge (\forall z \: z \mid w \Rightarrow ((w=z) \vee (z=1)))]$ Here $a \mid b$ ... $S3:$ Set of all integers Which of the above sets satisfy $\varphi$? S1 and S2 S1 and S3 S2 and S3 S1, S2 and S3
28
Let $m$ and $n$ range over natural numbers and let $\text{Prime}(n)$ be true if $n$ is a prime number. Which of the following formulas expresses the fact that the set of prime numbers is infinite? $(\forall m) (\exists n) (n > m) \text{ implies Prime}(n)$ ... $(\exists n) (\forall m) (n > m) \wedge \text{Prime}(n)$
29
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
30
Which one of the following predicate formulae is NOT logically valid? Note that $W$ is a predicate formula without any free occurrence of $x$. $\forall x (p(x) \vee W) \equiv \forall x \: ( px) \vee W$ $\exists x(p(x) \wedge W) \equiv \exists x \: p(x) \wedge W$ ... $\exists x(p(x) \rightarrow W) \equiv \forall x \: p(x) \rightarrow W$
31
Let $\{f_n(x)\}$ be a sequence of polynomials defined inductively as $f_1(x) = (x-2)^2$ $f_{n+1}(x) = (f_n(x)-2)^2, \qquad n\geq 1.$ Let $a_n$ and $b_n$ respectively denote the constant term and the coefficient of $x$ in $f_n(x).$ Then $a_n = 4, b_n = -4^n$ $a_n = 4, b_n = -4n^2$ $a_n = 4^{(n-1)!}, b_n = -4^n$ $a_n = 4^{(n-1)!}, b_n = -4n^2$
32
Which one of the following is the most appropriate logical formula to represent the statement? "Gold and silver ornaments are precious". The following notations are used: $G(x): x$ is a gold ornament $S(x): x$ is a silver ornament $P(x): x$ is precious $\forall x(P(x) \implies (G(x) \wedge S(x)))$ ... $\exists x((G(x) \wedge S(x)) \implies P(x))$ $\forall x((G(x) \vee S(x)) \implies P(x))$
33
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 some person at some 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
34
What is the logical translation of the following statement? "None of my friends are perfect." $∃x(F (x)∧ ¬P(x))$ $∃ x(¬ F (x)∧ P(x))$ $∃x(¬F (x)∧¬P(x))$ $¬∃ x(F (x)∧ P(x))$
35
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
36
For each fo these statements find a domain for which the statements is true and a domain for which the statement is false. Everyone is studying discrete mathematics. Everyone is older than 21 years. Everyone two people have the same mother. No two different people have the same grandmother.
37
Show that if $n$ is an integer and $n^3+5$ is odd, then $n$ is even using. a proof by contraposition. a proof by contradiction.
Prove that if $m$ and $n$ are integers and $mn$ is even, then $m$ is even or $n$ is even.
Show that the formula $\left[(\sim p \vee q) \Rightarrow (q \Rightarrow p)\right]$ is not a tautology. Let $A$ be a tautology and $B$ any other formula. Prove that $(A \vee B)$ is a tautology.
Which of the following is FALSE? $Read\ \wedge as\ AND, \vee\ as\ OR, \sim as\ NOT, \rightarrow$ as one way implication and $\leftrightarrow$ as two way implication? $((x\rightarrow y)\wedge x)\rightarrow y$ $((\sim x\rightarrow y)\wedge (\sim x\wedge \sim y))\rightarrow x$ $(x\rightarrow (x\vee y))$ $((x\vee y)\leftrightarrow (\sim x\vee \sim y))$