# Recent questions and answers in Mathematical Logic

1
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 ?
2
In the following, $A$ stands for a set of apples, and $S(x, y)$ stands for "$x$ is sweeter than $y$. Let $\Psi \equiv \exists x : x \in A$ $\Phi \equiv \forall x \in A : \exists y \in A : S(x, y).$ ...
3
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).
4
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
5
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? $(\forall m) (\exists n) (n > m) \text{ implies Prime}(n)$ ... $(\exists n) (\forall m) (n > m) \wedge \text{Prime}(n)$
6
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
7
Which one of the following predicate formulae is NOT logically valid? Note that $W$ is a predicate formula without any free occurrence of $x$. $\forall x (p(x) \vee W) \equiv \forall x \: ( px) \vee W$ $\exists x(p(x) \wedge W) \equiv \exists x \: p(x) \wedge W$ ... $\exists x(p(x) \rightarrow W) \equiv \forall x \: p(x) \rightarrow W$
8
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 $a_n = 4, b_n = -4^n$ $a_n = 4, b_n = -4n^2$ $a_n = 4^{(n-1)!}, b_n = -4^n$ $a_n = 4^{(n-1)!}, b_n = -4n^2$
9
Which one of the following propositional logic formulas is TRUE when exactly two of $p,q$ and $r$ are TRUE? $(( p \leftrightarrow q) \wedge r) \vee (p \wedge q \wedge \sim r)$ $( \sim (p \leftrightarrow q) \wedge r)\vee (p \wedge q \wedge \sim r)$ $( (p \to q) \wedge r) \vee (p \wedge q \wedge \sim r)$ $(\sim (p \leftrightarrow q) \wedge r) \wedge (p \wedge q \wedge \sim r)$
10
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 $\forall x(P(x) \implies (G(x) \wedge S(x)))$ ... $\exists x((G(x) \wedge S(x)) \implies P(x))$ $\forall x((G(x) \vee S(x)) \implies P(x))$
11
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
12
What is the logical translation of the following statement? "None of my friends are perfect." $∃x(F (x)∧ ¬P(x))$ $∃ x(¬ F (x)∧ P(x))$ $∃x(¬F (x)∧¬P(x))$ $¬∃ x(F (x)∧ P(x))$
13
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
14
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.
15
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.
16
Prove that if $m$ and $n$ are integers and $mn$ is even, then $m$ is even or $n$ is even.
17
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.
18
"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) $(X\land \neg Z) \rightarrow Y$ $(X \land Y) \rightarrow \neg Z$ $X \rightarrow(Y\land \neg Z)$ $(X \rightarrow Y)\land \neg Z$
19
The binary operation $\Box$ ... $P \vee Q$? $\neg Q \Box \neg P$ $P\Box \neg Q$ $\neg P\Box Q$ $\neg P\Box \neg Q$
20
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
1 vote
21
Which of the following statements is false? $(P\land Q)\lor(\sim P\land Q)\lor(P \land \sim Q)$ is equal to $\sim Q\land \sim P$ $(P\land Q)\lor(\sim P\land Q)\lor(P \wedge \sim Q)$ is equal to $Q\lor P$ $(P\wedge Q)\lor (\sim P\land Q)\lor(P \wedge \sim Q)$ is equal to $Q\lor (P\wedge \sim Q)$ $(P\land Q)\lor(\sim P\land Q)\lor (P \land \sim Q)$ is equal to $P\lor (Q\land \sim P)$
22
Consider three vectors $x=\begin{bmatrix}1\\2 \end{bmatrix}, y=\begin{bmatrix}4\\8 \end{bmatrix},z=\begin{bmatrix}3\\1 \end{bmatrix}$. Which of the folowing statements is true $x$ and $y$ are linearly independent $x$ and $y$ are linearly dependent $x$ and $z$ are linearly dependent $y$ and $z$ are linearly dependent
23
A ladder $13$ feet long rests against the side of a house. The bottom of the ladder slides away from the house at a rate of $0.5$ ft/s. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is $5$ feet from the house? $\dfrac{5}{24} \text{ ft/s} \\$ $\dfrac{5}{12} \text{ ft/s} \\$ $-\dfrac{5}{24} \text {ft/s} \\$ $-\dfrac{5}{12} \text{ ft/s}$
1 vote
24
A traffic office imposes on an average 5 number of penalities daily on traffic violators.Assume that the number of the penalties on different days is independent and follows a Poisson distribution.The probability that there will be less than 4 penalties in a day is _______________ ?
25
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
1 vote
26
1 vote
27
A binary relation R on Z × Z is defined as follows: (a, b) R (c, d) iff a = c or b = d Consider the following propositions: 1. R is reflexive. 2. R is symmetric. 3. R is antisymmetric. Which one of the above statements is True?
28
Which one of the following well-formed formulae is a tautology? $\forall x \, \exists y \, R(x,y) \, \leftrightarrow \, \exists y \, \forall x \, R(x, y)$ $( \forall x \, [\exists y \, R(x,y) \, \rightarrow \, S(x, y)]) \, \rightarrow \, \forall x \, \exists y \, S(x, y)$ ... $\forall x \, \forall y \, P(x,y) \, \rightarrow \, \forall x \, \forall y \, P(y, x)$
29
Symbolize the expression "Every mother loves her children" in predicate logic.
30
The following propositional statement is $\left(P \implies \left(Q \vee R\right)\right) \implies \left(\left(P \wedge Q \right)\implies R\right)$ satisfiable but not valid valid a contradiction None of the above
31
Which one of the following well-formed formulae in predicate calculus is NOT valid ? $(\forall _{x} p(x) \implies \forall _{x} q(x)) \implies (\exists _{x} \neg p(x) \vee \forall _{x} q(x))$ $(\exists x p(x) \vee \exists x q (x)) \implies \exists x (p(x) \vee q (x))$ ... $\forall x (p(x) \vee q(x)) \implies (\forall x p(x) \vee \forall x q(x))$
32
Consider the following first order formula: ... Does it have finite models? Is it satisfiable? If so, give a countable model for it.
33
1 vote
34
Choose the most appropriate option. The Newton-Raphson iteration $x_{n+1}=\dfrac{x_{n}}{2}+\dfrac{3}{2x_{n}}$ can be used to solve the equation $x^{2}=3$ $x^{3}=3$ $x^{2}=2$ $x^{3}=2$
35
If $A$ and $B$ are two related events, and $P(A \mid B)$ represents the conditional probability, Bayes’ theorem states that $P(A\mid B) = \dfrac{P(A)}{P(B)} P(B\mid A)$ $P(A\mid B) = P(A) P(B) P(B\mid A)$ $P(A\mid B) = \dfrac{P(A)}{P(B)}$ $P(A\mid B) = P(A)+P(B)$
1 vote
36
If $y=f(x)$, in the interval $[a,b]$ is rotated about the $x$-axis, the Volume of the solid of revolution is $(f’(x)=dy/dx)$ $\int_{a}^{b} \pi [f(x)]^{2} dx \\$ $\int_{a}^{b}[f(x)]^{3} dx \\$ $\int_{a}^{b} \pi [{f}'(x)]^{2} dx \\$ $\int_{a}^{b} \pi^{2} f(x)dx \\$
37
The area under the curve $y(x)=3e^{-5x}$ from $x=0 \text{ to } x=\infty$ is $\dfrac{3}{5}$ $\dfrac{-3}{5}$ ${5}$ $\dfrac{5}{3}$
The eigenvalues of the matrix $\begin{bmatrix}1 & 2\\ 4 & 3 \end{bmatrix}$ are $\text{5 and -5}$ $\text{5 and -1}$ $\text{1 and -5}$ $\text{2 and 3}$
$\underset{x \rightarrow 0}{\lim} \dfrac{x^{3}+x^{2}-5x-2}{2x^{3}-7x^{2}+4x+4}=?$ $-0.5$ $(0.5)$ $\infty$ None of the above
$\displaystyle \int_{0}^{\dfrac{\pi}{2}} \sin^{7}\theta \cos ^{4} \theta d\theta=?$ $16/1155$ $16/385$ $16\pi/385$ $8\pi/385$