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Recent questions and answers in Engineering Mathematics

38 votes
5 answers
1
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) $((∀x)[α] ⇒ (∀x)[β]) ⇒ (∀x)[α ⇒ β]$ $(∀x)[α] ⇒ (∃x)[α ∧ β]$ $((∀x)[α ∨ β] ⇒ (∃x)[α]) ⇒ (∀x)[α]$ $(∀x)[α ⇒ β] ⇒ (((∀x)[α]) ⇒ (∀x)[β])$
answered 12 hours ago in Mathematical Logic tirth_patel 10k views
22 votes
5 answers
2
Which of the following statements is FALSE? The set of rational numbers is an abelian group under addition The set of integers in an abelian group under addition The set of rational numbers form an abelian group under multiplication The set of real numbers excluding zero is an abelian group under multiplication
answered 1 day ago in Set Theory & Algebra endurance1 7.7k views
1 vote
2 answers
3
Let P(x) , Q(x) and R(x) be the statements "x is a clear explantion " , "x is satisfactory " and "x is an excuse " resp Suppose that the domain x consists of all English text Express each of these using quantifiers , logical connectives and p(x) ... ) 1) All clear explanation are satisfactory 2) Some excuses are unsatisfactory 3) Some excuses are not clear explanation 4) does 3 follow from 1and 2
answered 2 days ago in Mathematical Logic ijnuhb 124 views
22 votes
5 answers
4
Choose the correct alternatives (more than one may be correct) and write the corresponding letters only: If $F_1$, $F_2$ and $F_3$ are propositional formulae such that $F_1 \land F_2 \rightarrow F_3$ and $F_1 \land F_2 \rightarrow \sim F_3$ are both ... and $F_2$ are tautologies The conjunction $F_1 \land F_2$ is not satisfiable Neither is tautologous Neither is satisfiable None of the above.
answered 3 days ago in Mathematical Logic subbus 3.2k views
0 votes
3 answers
5
Let $G=(V, E)$ be an undirected unweighted connected graph. The diameter of $G$ is defined as: diam$(G)=\displaystyle \max_{u,v\in V} \{$the length of shortest path between $u$ and $v\}$ Let $M$ be the adjacency matrix of $G$. Define graph $G_2$ on the same set of vertices with adjacency ... diam$(G)/2\rceil<$diam$(G_2)<$ diam$(G)$ diam$(G_2)$ = diam$(G)$ diam$(G)<$ diam$(G_2)\leq 2 $ diam$(G)$
answered 4 days ago in Graph Theory ascend 224 views
20 votes
5 answers
6
Choose the correct alternatives (More than one may be correct). Indicate which of the following well-formed formulae are valid: $\left(P\Rightarrow Q\right) {\wedge} \left(Q \Rightarrow R\right) \Rightarrow \left(P \Rightarrow R\right)$ ...
answered 5 days ago in Mathematical Logic subbus 3.7k views
2 votes
4 answers
7
A relation $R$ is said to be circular if $aRb$ and $bRc$ together imply $cRa$. Which of the following options is/are correct? If a relation $S$ is reflexive and symmetric, then $S$ is an equivalence relation. If a relation $S$ is circular and symmetric, ... and circular, then $S$ is an equivalence relation. If a relation $S$ is transitive and circular, then $S$ is an equivalence relation.
answered 5 days ago in Set Theory & Algebra Vallabh Mandare 357 views
10 votes
7 answers
8
Answer the following: Which of the following well-formed formulas are equivalent? $P \rightarrow Q$ $\neg Q \rightarrow \neg P$ $\neg P \vee Q$ $\neg Q \rightarrow P$
answered 5 days ago in Mathematical Logic subbus 1.4k views
13 votes
6 answers
9
Show that the conclusion $(r \to q)$ follows from the premises: $p, (p \to q) \vee (p \wedge (r \to q))$
answered 6 days ago in Mathematical Logic subbus 1.5k views
0 votes
2 answers
10
Consider the following expression. $\displaystyle \lim_{x\rightarrow-3}\frac{\sqrt{2x+22}-4}{x+3}$ The value of the above expression (rounded to 2 ddecimal places) is ___________.
answered 6 days ago in Calculus Harshq 244 views
0 votes
4 answers
11
Let $p$ and $q$ be two propositions. Consider the following two formulae in propositional logic. $S_1: (\neg p\wedge(p\vee q))\rightarrow q$ $S_2: q\rightarrow(\neg p\wedge(p\vee q))$ Which one of the following choices is correct? Both $S_1$ and $S_2$ are tautologies ... tautology but $S_2$ is not a tautology $S_1$ is not a tautology but $S_2$ is a tautology Niether $S_1$ nor $S_2$ is a tautology
answered 6 days ago in Mathematical Logic Harshq 204 views
0 votes
2 answers
12
Given a grid of $4\times4$ points,how many triangles with their vertices on the grid can be drawn?
answered Feb 24 in Combinatory reboot 350 views
21 votes
3 answers
13
0 votes
2 answers
14
2 votes
2 answers
15
If $\begin{vmatrix} 10! & 11! & 12! \\ 11! & 12! & 13! \\ 12! & 13! & 14! \end{vmatrix} = k(10!)(11!)(12!)$, then the value of $k$ is $1$ $2$ $3$ $4$
answered Feb 23 in Linear Algebra Auditi 111 views
1 vote
1 answer
16
For a statement $S$ in a program, in the context of liveness analysis, the following sets are defined: $\text{USE(S)}$ : the set of variables used in $S$ $\text{IN(S)}$ : the set of variables that are live at the entry of $S$ $\text{OUT(S)}$ : the set of variables that are live at the exit ... $) }\cup \text{ OUT ($S_2$)}$ $\text{OUT ($S_1$)}$ = $\text{USE ($S_1$)} \cup \text{IN ($S_2$)}$
answered Feb 22 in Linear Algebra swettt871 241 views
3 votes
4 answers
17
A bag has $r$ red balls and $b$ black balls. All balls are identical except for their colours. In a trial, a ball is randomly drawn from the bag, its colour is noted and the ball is placed back into the bag along with another ball of the same colour. Note that the number of balls in the bag will ...
answered Feb 21 in Probability Exynos 717 views
5 votes
4 answers
18
Match List-I with List-II: ... - (iv) (a) - (iv); (b) - (i); (c) - (iii); (d) - (ii) (a) - (iv); (b) - (iii); (c) - (i); (d) - (ii)
answered Feb 20 in Mathematical Logic anjli 769 views
0 votes
2 answers
19
In propositional logic if $\left ( P \rightarrow Q \right )\wedge \left ( R \rightarrow S \right )$ and $\left ( P \vee R \right )$ are two premises such that $\begin{array}{c} (P \to Q) \wedge (R \to S) \\ P \vee R \\ \hline Y \\ \hline \end{array}$ $Y$ is the premise : $P \vee R$ $P \vee S$ $Q \vee R$ $Q \vee S$
answered Feb 20 in Mathematical Logic anjli 329 views
0 votes
3 answers
21
1 vote
3 answers
22
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 $P\lor (Q\land \sim P)$
answered Feb 20 in Mathematical Logic anjli 183 views
0 votes
4 answers
23
0 votes
2 answers
24
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? $((x\rightarrow y)\wedge x)\rightarrow y$ $((\sim x\rightarrow y)\wedge (\sim x\wedge \sim y))\rightarrow x$ $(x\rightarrow (x\vee y))$ $((x\vee y)\leftrightarrow (\sim x\vee \sim y))$
answered Feb 20 in Mathematical Logic anjli 127 views
0 votes
2 answers
25
38 votes
12 answers
26
1 vote
1 answer
27
The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter $2$. For a randomly picked component of this type, the probability that its lifetime exceeds the expected lifetime (rounded to $2$ decimal places) is ____________.
answered Feb 19 in Probability jatinmittal199510 266 views
1 vote
2 answers
28
Which of the following pairs of propositions are not logically equivalent? $((p \rightarrow r) \wedge (q \rightarrow r))$ and $((p \vee q) \rightarrow r)$ $p \leftrightarrow q$ and $(\neg p \leftrightarrow \neg q)$ $((p \wedge q) \vee (\neg p \wedge \neg q))$ and $p \leftrightarrow q$ $((p \wedge q) \rightarrow r)$ and $((p \rightarrow r) \wedge (q \rightarrow r))$
answered Feb 19 in Discrete Mathematics anjli 238 views
3 votes
3 answers
29
How many partial functions are there from a set with m elements to a set with n elements? Q. I cannot get the intuition how the solution arrived to be (n+1)^m
answered Feb 19 in Set Theory & Algebra 5a1n1amarjeet 957 views
2 votes
1 answer
30
There are $6$ jobs with distinct difficulty levels, and $3$ computers with distinct processing speeds. Each job is assigned to a computer such that: The fastest computer gets the toughest job and the slowest computer gets the easiest job. Every computer gets at least one job. The number of ways in which this can be done is ___________.
answered Feb 19 in Combinatory Arjun 505 views
1 vote
2 answers
31
Let $S$ be a set of consisting of $10$ elements. The number of tuples of the form $(A,B)$ such that $A$ and $B$ are subsets of $S$, and $A \subseteq B$ is ___________
answered Feb 19 in Set Theory & Algebra Shaik Masthan 382 views
2 votes
2 answers
32
Choose the correct choice(s) regarding the following proportional logic assertion $S$: $S: (( P \wedge Q) \rightarrow R) \rightarrow (( P \wedge Q) \rightarrow (Q \rightarrow R))$ $S$ is neither a tautology nor a contradiction $S$ is a tautology $S$ is a contradiction The antecedent of $S$ is logically equivalent to the consequent of $S$
answered Feb 19 in Mathematical Logic Shaik Masthan 346 views
2 votes
2 answers
33
Let $G$ be a group of order $6$, and $H$ be a subgroup of $G$ such that $1<|H|<6$. Which one of the following options is correct? Both $G$ and $H$ are always cyclic $G$ may not be cyclic, but $H$ is always cyclic $G$ is always cyclic, but $H$ may not be cyclic Both $G$ and $H$ may not be cyclic
answered Feb 18 in Set Theory & Algebra Sanjay Sharma 218 views
2 votes
1 answer
34
Suppose that $P$ is a $4 \times 5$ matrix such that every solution of the equation $\text{Px=0}$ is a scalar multiple of $\begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T$. The rank of $P$ is __________
answered Feb 18 in Linear Algebra Gaurav Parashar 295 views
1 vote
2 answers
35
In an examination, a student can choose the order in which two questions ($\textsf{QuesA}$ and $\textsf{QuesB}$) must be attempted. If the first question is answered wrong, the student gets zero marks. If the first question is answered correctly and the second question ... and then $\textsf{QuesA}$. Expected marks $22$. First $\textsf{QuesA}$ and then $\textsf{QuesB}$. Expected marks $16$.
answered Feb 18 in Probability jatinmittal199510 460 views
1 vote
1 answer
36
Consider the following sets, where $n \geq 2$: $S_1$: Set of all $n \times n$ matrices with entries from the set $\{ a, b, c\}$ $S_2$: Set of all functions from the set $\{0,1,2, \dots, n^2-1\}$ to the set $\{0, 1, 2\}$ Which of the following ... to $S_2$ There exists a surjection from $S_1$ to $S_2$ There exists a bijection from $S_1$ to $S_2$ There does not exist an injection from $S_1$ to $S_2$
asked Feb 18 in Set Theory & Algebra Arjun 339 views
2 votes
1 answer
37
For a given biased coin, the probability that the outcome of a toss is a head is $0.4$. This coin is tossed $1,000$ times. Let $X$ denote the random variable whose value is the number of times that head appeared in these $1,000$ tosses. The standard deviation of $X$ (rounded to $2$ decimal place) is _________
asked Feb 18 in Probability Arjun 256 views
2 votes
2 answers
38
Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function on the interval $[-3, 3]$ and a differentiable function in the interval $(-3,3)$ such that for every $x$ in the interval, $f’(x) \leq 2$. If $f(-3)=7$, then $f(3)$ is at most __________
asked Feb 18 in Calculus Arjun 229 views
1 vote
1 answer
39
For two $n$-dimensional real vectors $P$ and $Q$, the operation $s(P,Q)$ is defined as follows: $s(P,Q) = \displaystyle \sum_{i=1}^n (P[i] \cdot Q[i])$ Let $\mathcal{L}$ be a set of $10$-dimensional non-zero real vectors such that for every pair of distinct vectors $P,Q \in \mathcal{L}$, $s(P,Q)=0$. What is the maximum cardinality possible for the set $\mathcal{L}$? $9$ $10$ $11$ $100$
asked Feb 18 in Set Theory & Algebra Arjun 221 views
0 votes
1 answer
40
Consider the two statements. $S_1$: There exist random variables $X$ and $Y$ such that $ \\ \left(\mathbb E[(X-\mathbb E(X))(Y-\mathbb E(Y))]\right)^2>\textsf{Var}[X]\textsf{Var}[Y]$ $S_2$: For all random variables $X$ ... ? Both $S_1$ and $S_2$ are true $S_1$ is true, but $S_2$ is false $S_1$ is false, but $S_2$ is true Both $S_1$ and $S_2$ are false
asked Feb 18 in Probability Arjun 223 views
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