Recent questions in Discrete Mathematics

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1

UGCNET-Dec2007-II: 2
The number of edges in a complete graph with ‘n’ vertices is equal to : $n(n-1)$ $\large\frac{n(n-1)}{2}$ $n^2$ $2n-1$

asked 1 day ago in Graph Theory by jothee Veteran (106k points) | 10 views

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0 answers

2

UGCNET-Dec2006-II: 2
The proposition ~ q ∨ p is equivalent to :

asked 1 day ago in Mathematical Logic by jothee Veteran (106k points) | 14 views

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1 answer

3

UGCNET-Dec2006-II: 3
The number of edges in a complete graph with N vertices is equal to: $N (N−1)$ $2N−1$ $N−1$ $N(N−1)/2$

asked 1 day ago in Graph Theory by jothee Veteran (106k points) | 10 views

0 votes

1 answer

4

UGCNET-Dec2004-II: 2
If $f (x)=x+1\:\text{and}\:g(x)=x+3$ then $f 0 f 0 f 0 f$ is : $g$ $g+1$ $g^4$ None of these

asked 3 days ago in Set Theory & Algebra by jothee Veteran (106k points) | 7 views

0 votes

1 answer

5

UGCNET-Dec2004-II: 4
The following lists are the degrees of all the vertices of a graph : $1,2,3,4,5$ $3,4,5,6,7$ $1, 4, 5, 8, 6$ $3,4,5,6$ (i) and (ii) (iii) and (iv) (iii) and (ii) (ii) and (iv)

asked 3 days ago in Graph Theory by jothee Veteran (106k points) | 11 views

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6

UGCNET-Dec2004-II: 23
Weighted graph : Is a bi-directional graph. Is directed graph. Is graph in which number associated with arc. Eliminates table method.

asked 3 days ago in Graph Theory by jothee Veteran (106k points) | 4 views

0 votes

1 answer

7

JEST 2020
X AND Y is an arbitrary sets, F: $X\rightarrow Y$ show that a and b are equivalent F is one-one For all set Z and function g1: $Z\rightarrow X$ and g2: $Z\rightarrow X$, if $g1 \neq g2$ implies $f \bigcirc g1 \neq f \bigcirc g2$ Where $\bigcirc$ is a fucntion composition.

asked Feb 17 in Set Theory & Algebra by vivek_mishra Junior (653 points) | 150 views

+4 votes

4 answers

8

GATE2020-CS-1
Consider the functions $e^{-x}$ $x^{2}-\sin x$ $\sqrt{x^{3}+1}$ Which of the above functions is/are increasing everywhere in $[ 0,1]$? Ⅲ only Ⅱ only Ⅱ and Ⅲ only Ⅰ and Ⅲ only

asked Feb 12 in Mathematical Logic by Arjun Veteran (435k points) | 1.6k views

+3 votes

4 answers

9

GATE2020-CS-39
Which one of the following predicate formulae is NOT logically valid? Note that $W$ is a predicate formula without any free occurrence of $x$. $\forall x (p(x) \vee W) \equiv \forall x \: ( px) \vee W$ ... $\exists x(p(x) \rightarrow W) \equiv \forall x \: p(x) \rightarrow W$

asked Feb 12 in Mathematical Logic by Arjun Veteran (435k points) | 1.9k views

+3 votes

6 answers

10

GATE2020-CS-42
The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ______.

asked Feb 12 in Combinatory by Arjun Veteran (435k points) | 1.4k views

+2 votes

3 answers

11

GATE2020-CS-45
For $n>2$, let $a \in \{0,1\}^n$ be a non-zero vector. Suppose that $x$ is chosen uniformly at random from $\{0,1\}^n$. Then, the probability that $\displaystyle{} \Sigma_{i=1}^n a_i x_i$ is an odd number is______________

asked Feb 12 in Mathematical Logic by Arjun Veteran (435k points) | 985 views

+2 votes

3 answers

12

GATE2020-CS-52
Graph $G$ is obtained by adding vertex $s$ to $K_{3,4}$ and making $s$ adjacent to every vertex of $K_{3,4}$. The minimum number of colours required to edge-colour $G$ is _______

asked Feb 12 in Graph Theory by Arjun Veteran (435k points) | 1.2k views

0 votes

3 answers

13

TIFR2020-B-11
Which of the following graphs are bipartite? Only $(1)$ Only $(2)$ Only $(2)$ and $(3)$ None of $(1),(2),(3)$ All of $(1),(2),(3)$

asked Feb 11 in Graph Theory by Lakshman Patel RJIT Veteran (61.2k points) | 89 views

+3 votes

2 answers

14

ISRO2020-73
Given that $B(a)$ means “$a$ is a bear” $F(a)$ means “$a$ is a fish” and $E(a,b)$ means “$a $ eats $b$” Then what is the best meaning of $\forall x [F(x) \to \forall y(E(y,x)\rightarrow b(y))]$ Every fish is eaten by some bear Bears eat only fish Every bear eats fish Only bears eat fish

asked Jan 13 in Mathematical Logic by Satbir Boss (25.3k points) | 368 views

+1 vote

1 answer

15

ISRO2020-76
If $A=\{x,y,z\}$ and $B=\{u,v,w,x\}, $ and the universe is $\{s,t,u,v,w,x,y,z\}$ Then $(A \cup \bar{B}) \cap (A \cap B)$ is equal to $\{u,v,w,x\}$ $\{ \ \}$ $\{u,v,w,x,y,z\}$ $\{u,v,w\}$

asked Jan 13 in Set Theory & Algebra by Satbir Boss (25.3k points) | 192 views

0 votes

1 answer

16

$\textbf{NTA NET DEC 2019 (group)}$
Consider the following statements: $\mathbf{S_1:}\;\;$If a group $\mathbf{(G,*)}$ is of order $\mathbf n$ and $\mathrm {a \in G}$ is such that $\mathrm {a^m=e}$ for some integer $\mathrm {m \le n}$ then $\mathbf m$ must divide $\mathbf n$ ... $(3)\;\;\;\text{Niether}\; \mathrm{S_1}\;\text{nor}\;\mathrm{S_2}$

asked Dec 29, 2019 in Set Theory & Algebra by Sanjay Sharma Boss (49.5k points) | 140 views

+1 vote

1 answer

17

NTA NET DEC2019( shortest distance)
Consider a weighted directed graph. The current shortest distance from sources S to node x is represented by d[v] = 29 . d[u] = 15 , w[u,v] = 12. What is the updated value of d[v]based on current information? 1) 29 2) 27 3) 25 4) 17

asked Dec 22, 2019 in Graph Theory by Sanjay Sharma Boss (49.5k points) | 147 views

+3 votes

0 answers

18

The Interesting combination sum problems
Find the number of possible solutions for $x,y,z$ for each the following cases. $Case\ 1.$ Case of unlimited repetition. $x + y +z = 10$ and $x \geq 0\ , y \geq 0,\ z \geq 0 $ $Case\ 2 $ Case of unlimited repetition with variable lower bounds $x + y +z = 10$ and ... variable. $x + y +z = 10$ and $8 \geq x \geq 1\ , \ 20 \geq y \geq 2 \ , 12 \geq z \geq 3\ $

asked Nov 1, 2019 in Combinatory by Satbir Boss (25.3k points) | 244 views

+2 votes

2 answers

19

ISI2014-DCG-1
Let $(1+x)^n = C_0+C_1x+C_2x^2+ \dots + C_nx^n$, $n$ being a positive integer. The value of $\left( 1+\dfrac{C_0}{C_1} \right) \left( 1+\dfrac{C_1}{C_2} \right) \cdots \left( 1+\dfrac{C_{n-1}}{C_n} \right)$ is $\left( \frac{n+1}{n+2} \right) ^n$ $ \frac{n^n}{n!} $ $\left( \frac{n}{n+1} \right) ^n$ $ \frac{(n+1)^n}{n!} $

asked Sep 23, 2019 in Combinatory by Arjun Veteran (435k points) | 202 views

+1 vote

1 answer

20

ISI2014-DCG-5
Consider the sets defined by the real solutions of the inequalities $A = \{(x,y):x^2+y^4 \leq 1\} \:\:\:\:\:\:\: B=\{(x,y):x^4+y^6 \leq 1\}$ Then $B \subseteq A$ $A \subseteq B$ Each of the sets $A – B, \: B – A$ and $A \cap B$ is non-empty none of the above

asked Sep 23, 2019 in Set Theory & Algebra by Arjun Veteran (435k points) | 115 views

+2 votes

1 answer

21

ISI2014-DCG-15
Let $\mathbb{N}=\{1,2,3, \dots\}$ be the set of natural numbers. For each $n \in \mathbb{N}$, define $A_n=\{(n+1)k, \: k \in \mathbb{N} \}$. Then $A_1 \cap A_2$ equals $A_3$ $A_4$ $A_5$ $A_6$

asked Sep 23, 2019 in Set Theory & Algebra by Arjun Veteran (435k points) | 59 views

+2 votes

3 answers

22

ISI2014-DCG-18
$^nC_0+2^nC_1+3^nC_2+\cdots+(n+1)^nC_n$ equals $2^n+n2^{n-1}$ $2^n-n2^{n-1}$ $2^n$ none of these

asked Sep 23, 2019 in Combinatory by Arjun Veteran (435k points) | 99 views

+2 votes

0 answers

23

ISI2014-DCG-32
Consider $30$ multiple-choice questions, each with four options of which exactly one is correct. Then the number of ways one can get only the alternate questions correctly answered is $3^{15}$ $2^{31}$ $2 \times \begin{pmatrix} 30 \\ 15 \end{pmatrix}$ $2 \times 3^{15}$

asked Sep 23, 2019 in Combinatory by Arjun Veteran (435k points) | 110 views

+1 vote

1 answer

24

ISI2014-DCG-34
The following sum of $n+1$ terms $2 + 3 \times \begin{pmatrix} n \\ 1 \end{pmatrix} + 5 \times \begin{pmatrix} n \\ 2 \end{pmatrix} + 9 \times \begin{pmatrix} n \\ 3 \end{pmatrix} + 17 \times \begin{pmatrix} n \\ 4 \end{pmatrix} + \cdots$ up to $n+1$ terms is equal to $3^{n+1}+2^{n+1}$ $3^n \times 2^n$ $3^n + 2^n$ $2 \times 3^n$

asked Sep 23, 2019 in Combinatory by Arjun Veteran (435k points) | 68 views

+1 vote

1 answer

25

ISI2014-DCG-35
Let $A$ and $B$ be disjoint sets containing $m$ and $n$ elements respectively, and let $C=A \cup B$. Then the number of subsets $S$ (of $C$) which contains $p$ elements and also has the property that $S \cap A$ contains $q$ ... $\begin{pmatrix} m \\ p-q \end{pmatrix} \times \begin{pmatrix} n \\ q \end{pmatrix}$

asked Sep 23, 2019 in Set Theory & Algebra by Arjun Veteran (435k points) | 55 views

+1 vote

1 answer

26

ISI2014-DCG-41
The number of permutations of the letters $a, b, c$ and $d$ such that $b$ does not follow $a,c$ does not follow $b$, and $c$ does not follow $d$, is $11$ $12$ $13$ $14$

asked Sep 23, 2019 in Combinatory by Arjun Veteran (435k points) | 69 views

+2 votes

1 answer

27

ISI2014-DCG-63
If $^nC_{r-1}=36$, $^nC_r=84$ an $^nC_{r+1}=126$ then $r$ is equal to $1$ $2$ $3$ none of these

asked Sep 23, 2019 in Combinatory by Arjun Veteran (435k points) | 48 views

+1 vote

1 answer

28

ISI2014-DCG-66
Consider all possible words obtained by arranging all the letters of the word $\textbf{AGAIN}$. These words are now arranged in the alphabetical order, as in a dictionary. The fiftieth word in this arrangement is $\text{IAANG}$ $\text{NAAGI}$ $\text{NAAIG}$ $\text{IAAGN}$

asked Sep 23, 2019 in Combinatory by Arjun Veteran (435k points) | 57 views

+1 vote

2 answers

29

ISI2014-DCG-71
Five letters $A, B, C, D$ and $E$ are arranged so that $A$ and $C$ are always adjacent to each other and $B$ and $E$ are never adjacent to each other. The total number of such arrangements is $24$ $16$ $12$ $32$

asked Sep 23, 2019 in Combinatory by Arjun Veteran (435k points) | 59 views

+1 vote

1 answer

30

ISI2014-DCG-72
The sum $\sum_{k=1}^n (-1)^k \:\: {}^nC_k \sum_{j=0}^k (-1)^j \: \: {}^kC_j$ is equal to $-1$ $0$ $1$ $2^n$

asked Sep 23, 2019 in Combinatory by Arjun Veteran (435k points) | 56 views

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