Questions by ankitgupta.1729 / GATE Overflow for GATE CSE

6 votes

0 answers

21

Discrete Mathematics | Propositional Logic | Test 1 | Question: 7
The $\textit{well-formed formulas (wff)}$ of propositional logic are obtained by using the following rules: 1. An atomic proposition $\phi$ is a well-formed formula. 2. If $\phi$ ... (P, Q and R are atomic propositions) Total number of well-formed formulas are ______

The $\textit{well-formed formulas (wff)}$ of propositional logic are obtained by using the following rules: 1. An atomic proposition $\phi$ is a well-formed formula. 2. ...

637 views

asked Apr 11, 2023

3 votes

1 answer

22

Discrete Mathematics | Propositional Logic | Test 1 | Question: 8
Consider the following truth table for the connective $\rightarrow:$ ... (i) and (iii) are correct (i) and (ii) are correct (i), (ii) and (iii) are correct

Consider the following truth table for the connective $\rightarrow:$ $$\begin{array}{c|c|c}p & q & p \rightarrow q \\\hlineT & T & T \\T & F & F \\F & T...

254 views

asked Apr 11, 2023

2 votes

0 answers

23

Discrete Mathematics | Propositional Logic | Test 1 | Question: 9
Consider the following statements: "Ralph is a dog if he's not a puppet" can be formalized as $\neg$ (Ralph is a puppet) $\rightarrow$ (Ralph is a dog) "Ralph is not a dog because he's a puppet" ... correct $(i)$ and $(iii)$ are correct $(i),(ii)$ and $(iii)$ are correct

Consider the following statements: "Ralph is a dog if he’s not a puppet" can be formalized as $\neg$ (Ralph is a puppet) $\rightarrow$ (Ralph is a dog) ...

390 views

asked Apr 11, 2023

3 votes

1 answer

24

Discrete Mathematics | Propositional Logic | Test 1 | Question: 10
Consider the following two statements: i. Sentence $\textit{Neither A nor B}$ can be represented by $A \downarrow B$ where $\downarrow$ is used in Boolean circuits for $\textit{nor}$ function. ii. Sentence $\textit{not at once A and B}$ ... $(i)$ is correct Only $(ii)$ is correct Both $(i)$ and $(ii)$ are correct None of the above

Consider the following two statements: i. Sentence $\textit{Neither A nor B}$ can be represented by $A \downarrow B$ where $\downarrow$ is used in Boolean circui...

196 views

asked Apr 11, 2023

5 votes

1 answer

25

Discrete Mathematics | Propositional Logic | Test 1 | Question: 11
A function $f:\{0,1\}^n \rightarrow \{0,1\}$ is called an $\textit{n-ary Boolean function}$ or $\textit{truth function}.$ We denote their totality by the set $\mathbf{B_n}.$ Now, $f \in \mathbf{B_n}$ is called $\textit{linear}$ ... of $\textit{n-ary linear Boolean functions}$ is: $2^{2^n}$ $2^{2^{n+1}}$ $2^n$ $2^{n+1}$

A function $f:\{0,1\}^n \rightarrow \{0,1\}$ is called an $\textit{n-ary Boolean function}$ or $\textit{truth function}.$ We denote their totality by the set $\m...

463 views

asked Apr 11, 2023

6 votes

0 answers

26

Discrete Mathematics | Propositional Logic | Test 1 | Question: 12
The atomic propositional variables $p_0,p_1,...$ are $\textit{formulas},$ called $\textit{prime formulas},$ also called $\textit{atomic}$ formulas, or simply $\textit{primes}.$ ... a DNF nor a CNF. $p \vee \neg (\neg p \wedge q)$ is either a DNF or a CNF.

The atomic propositional variables $p_0,p_1,...$ are $\textit{formulas},$ called $\textit{prime formulas},$ also called $\textit{atomic}$ formulas, or simply $\textit{pri...

531 views

asked Apr 11, 2023

2 votes

1 answer

27

Discrete Mathematics | Propositional Logic | Test 1 | Question: 13
The set of logical symbols of a propositional language is called the $\textit{logical signature}.$ A logical signature is called $\textit{functionally complete}$ if every Boolean function is representable by a formula in this ... $\{\rightarrow\}$ is $\textit{not}$ functionally complete.

The set of logical symbols of a propositional language is called the $\textit{logical signature}.$ A logical signature is called $\textit{functionally complete}$ if every...

293 views

asked Apr 11, 2023

4 votes

1 answer

28

Discrete Mathematics | Propositional Logic | Test 1 | Question: 14
A compound sentence is a $\textit{tautology}$ if it is true independently of the truth values of its component atomic sentences. A sentence is $\textit{atomic}$ if it contains no sentential connectives. Let $P,Q$ and ... $(P \leftrightarrow P) \leftrightarrow P$ is a tautology Number of correct statements are ______

A compound sentence is a $\textit{tautology}$ if it is true independently of the truth values of its component atomic sentences. A sentence is $\textit{atomic}$ if it con...

317 views

asked Apr 11, 2023

3 votes

2 answers

29

Discrete Mathematics | Propositional Logic | Test 1 | Question: 15
A compound sentence is a $\textit{tautology}$ if it is true independently of the truth values of its component atomic sentences. A sentence is $\textit{atomic}$ if it contains no sentential connectives. A sentence $P$ ... ) $\neg Q \rightarrow \neg P$ $Q \rightarrow P$ $P \rightarrow Q$ $\neg P \wedge Q$

A compound sentence is a $\textit{tautology}$ if it is true independently of the truth values of its component atomic sentences. A sentence is $\textit{atomic}$ if it con...

419 views

asked Apr 11, 2023

6 votes

0 answers

30

IISc CSA - Research Interview Question
Prove that the rank of the Adjacency Matrix which is associated with a $k-$ regular graph is $k.$

Prove that the rank of the Adjacency Matrix which is associated with a $k-$ regular graph is $k.$

686 views

asked May 22, 2019

1 votes

0 answers

31

ISI-PCB-2015-C8-a
Let $a_{n−1}a_{n−2}...a_0$ and $b_{n−1}b_{n−2}...b_0$ denote the $2's$ complement representation of two integers $A$ and $B$ respectively. Addition of $A$ and $B$ yields a sum $S=s_{n−1}s_{n−2}...s_0.$ The outgoing carry generated at the most ... $\oplus$ denotes the Boolean XOR operation. You may use the Boolean identity: $X+Y=X⊕Y⊕(XY)$ to prove your result.

Let $a_{n−1}a_{n−2}...a_0$ and $b_{n−1}b_{n−2}...b_0$ denote the $2’s$ complement representation of two integers $A$ and $B$ respectively. Addition of $A$ and $...

587 views

asked Mar 26, 2019

0 votes

0 answers

32

ISI-PCB-2015-C5
Consider three relations $R_1(\underline{X},Y,Z), R_2(\underline{M},N,P),$ and $R_3(\underline{N,X})$. The primary keys of the relations are underlined. The relations have $100,30,$ and $400$ tuples, respectively. The space requirements for different attributes ... execution of the join. For, (a), Order could be anything and min. cost =$100*30*400*$total size of all the attributes.

Consider three relations $R_1(\underline{X},Y,Z), R_2(\underline{M},N,P),$ and $R_3(\underline{N,X})$. The primary keys of the relations are underlined. The relations hav...

546 views

asked Mar 26, 2019

0 votes

1 answer

33

ISI-PCB-2015-C1-b
A $64000$-byte message is to be transmitted over a $2$-hop path in a store-and-forward packet-switching network. The network limits packets toa maximum size of $2032$ bytes including a $32$-byte header. The trans-mission lines in the network are error free and have a speed of $50$ ... answer as $1*3*(T_t+T_p) + \;31*T_t$ where $T_t=0.325\; ms$ and $T_p=3.333\; ms$. Please Confirm.

A $64000$-byte message is to be transmitted over a $2$-hop path in a store-and-forward packet-switching network. The network limits packets toa maximum size of $2032$ byt...

1.1k views

asked Mar 25, 2019

2 votes

1 answer

34

ISI-MMA 2019 Sample Questions-23
For $n \geq1$, Let $a_{n} = \frac{1}{2^{2}} + \frac{2}{3^{2}} +.....+ \frac{n}{(n+1)^{2}}$ and $b_{n} = c_{0} + c_{1}r + c_{2}r^{2}+.....+c_{n}r^{n},$ where$|c_{k}| \leq M$ for all integers $k$ ... not a Cauchy sequence (C) $\{a_n\}$ is not a Cauchy sequence but $\{b_n\}$ is a Cauchy sequence (D) neither $\{a_n\}$ nor $\{b_n\}$ is a Cauchy sequence.

For $n \geq1$, Let$a_{n} = \frac{1}{2^{2}} + \frac{2}{3^{2}} +.....+ \frac{n}{(n+1)^{2}}$ and $b_{n} = c_{0} + c_{1}r + c_{2}r^{2}+.....+c_{n}r^{n},$where$|c_{k}| \leq M$...

1.2k views

asked Mar 17, 2019

1 votes

1 answer

35

ISI MMA-2015
Let, $a_{n} \;=\; \left ( 1-\frac{1}{\sqrt{2}} \right ) ... \left ( 1- \frac{1}{\sqrt{n+1}} \right )$ , $n \geq 1$. Then $\lim_{n\rightarrow \infty } a_{n}$ (A) equals $1$ (B) does not exist (C) equals $\frac{1}{\sqrt{\pi }}$ (D) equals $0$

Let, $a_{n} \;=\; \left ( 1-\frac{1}{\sqrt{2}} \right ) ... \left ( 1- \frac{1}{\sqrt{n+1}} \right )$ , $n \geq 1$. Then $\lim_{n\rightarrow \infty } a_{n}$(A) equals $1$...

1.3k views

asked Feb 21, 2019

2 votes

1 answer

36

ISI MMA-2015
If two real polynomials $f(x)$ and $g(x)$ of degrees $m\;(\geq2)$ and $n\;(\geq1)$ respectively, satisfy $f(x^{2}+1) = f(x)g(x)$ $,$ for every $x\in \mathbb{R}$ , then (A) $f$ has exactly one real root $x_{0}$ such that $f'(x_{0}) \neq 0$ (B) $f$ has exactly one real root $x_{0}$ such that $f'(x_{0}) = 0$ (C) $f$ has $m$ distinct real roots (D) $f$ has no real root.

If two real polynomials $f(x)$ and $g(x)$ of degrees $m\;(\geq2)$ and $n\;(\geq1)$ respectively, satisfy $f(x^{2}+1)...

1.2k views

asked Feb 20, 2019

1 votes

1 answer

37

Definite Integral
$\displaystyle S = \int_{0}^{2\pi } \sqrt{4\cos^{2}t +\sin^{2}t} \, \, dt$ Please explain how to solve it.

$\displaystyle S = \int_{0}^{2\pi } \sqrt{4\cos^{2}t +\sin^{2}t} \, \, dt$Please explain how to solve it.

952 views

asked Jun 11, 2018

0 votes

0 answers

38

Gilbert Strang - Real Symmetric Matrices
Prove that :- For all real symmetric matrices , No. of positive Pivots = No . of positive eigenvalues Can anyone please give the formal mathematical proof for the above statement ?

Prove that :- For all real symmetric matrices , No. of positive Pivots = No . of positive eigenvaluesCan anyone please give the formal mathematical p...

654 views

asked Jun 2, 2018

0 votes

0 answers

39

Universality of Uniform
According to Universality of Uniform , We can get from the uniform distribution to the other distributions and also from other distributions back to the uniform distribution. Please explain how we would simulate from one distribution to other distribution ?

According to Universality of Uniform ,We can get from the uniform distribution to the other distributions and also from other distributions back to the uniform distributi...

735 views

asked May 6, 2018

1 votes

1 answer

40

Group Theory
Prove that :- Every infinite cyclic group is isomorphic to the infinite cyclic group of integers under addition.

Prove that :-Every infinite cyclic group is isomorphic to the infinite cyclic group of integers under addition.

1.3k views

asked May 1, 2018

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