Highest voted questions in Discrete Mathematics / GATE Overflow for GATE CSE

46 votes

6 answers

91

GATE CSE 2002 | Question: 1.4
The minimum number of colours required to colour the vertices of a cycle with $n$ nodes in such a way that no two adjacent nodes have the same colour is $2$ $3$ $4$ $n-2 \left \lfloor \frac{n}{2} \right \rfloor+2$

The minimum number of colours required to colour the vertices of a cycle with $n$ nodes in such a way that no two adjacent nodes have the same colour is$2$$3$$4$$n-2 \lef...

Kathleen

11.3k views

Kathleen asked Sep 15, 2014

46 votes

3 answers

92

GATE CSE 2013 | Question: 47
Which one of the following is NOT logically equivalent to $¬∃x(∀ y (α)∧∀z(β ))$ ? $∀ x(∃ z(¬β )→∀ y(α))$ $∀x(∀ z(β )→∃ y(¬α))$ $∀x(∀ y(α)→∃z(¬β ))$ $∀x(∃ y(¬α)→∃z(¬β ))$

Which one of the following is NOT logically equivalent to $¬∃x(∀ y (α)∧∀z(β ))$ ?$∀ x(∃ z(¬β )→∀ y(α))$$∀x(∀ z(β )→∃ y(¬α))$$∀x(∀ y(�...

gatecse

12.0k views

gatecse asked Aug 21, 2014

45 votes

4 answers

93

GATE CSE 2016 Set 1 | Question: 2
Let $a_n$ be the number of $n$-bit strings that do NOT contain two consecutive $1's$. Which one of the following is the recurrence relation for $a_n$? $a_n = a_{n-1}+ 2a_{n-2}$ $a_n = a_{n-1}+ a_{n-2}$ $a_n = 2a_{n-1}+ a_{n-2}$ $a_n = 2a_{n-1}+ 2a_{n-2}$

Let $a_n$ be the number of $n$-bit strings that do NOT contain two consecutive $1's$. Which one of the following is the recurrence relation for $a_n$?$a_n = a_{n-1}+ 2a_{...

Sandeep Singh

9.6k views

Sandeep Singh asked Feb 12, 2016

45 votes

5 answers

94

GATE CSE 2015 Set 1 | Question: 26
$\sum\limits_{x=1}^{99}\frac{1}{x(x+1)}$ = ______.

$\sum\limits_{x=1}^{99}\frac{1}{x(x+1)}$ = ______.

makhdoom ghaya

8.2k views

makhdoom ghaya asked Feb 13, 2015

45 votes

10 answers

95

GATE IT 2005 | Question: 33
Let $A$ be a set with $n$ elements. Let $C$ be a collection of distinct subsets of $A$ such that for any two subsets $S_1$ and $S_2$ in $C$, either $S_1 \subset S_2$ or $S_2\subset S_1$. What is the maximum cardinality of $C?$ $n$ $n+1$ $2^{n-1} + 1$ $n!$

Let $A$ be a set with $n$ elements. Let $C$ be a collection of distinct subsets of $A$ such that for any two subsets $S_1$ and $S_2$ in $C$, either $S_1 \subset S_2$ or $...

Ishrat Jahan

11.9k views

Ishrat Jahan asked Nov 3, 2014

45 votes

4 answers

96

GATE IT 2006 | Question: 2
For the set $N$ of natural numbers and a binary operation $f : N \times N \to N,$ an element $z \in N$ is called an identity for $f,$ if $f (a, z) = a = f(z, a),$ for all $a \in N.$ Which of the following binary operations have an identity? $f (x, y) = x + y - 3$ $f (x, y) = \max(x, y)$ $f (x, y) = x^y$ I and II only II and III only I and III only None of these

For the set $N$ of natural numbers and a binary operation $f : N \times N \to N,$ an element $z \in N$ is called an identity for $f,$ if $f (a, z) = a = f(z, a),$ for all...

Ishrat Jahan

9.7k views

Ishrat Jahan asked Oct 30, 2014

45 votes

3 answers

97

GATE CSE 1997 | Question: 13
Let $F$ be the set of one-to-one functions from the set $\{1, 2, \dots, n\}$ to the set $\{1, 2,\dots, m\}$ where $m\geq n\geq1$. How many functions are members of $F$? How many functions $f$ in $F$ satisfy the property $f(i)=1$ for some $i, 1\leq i \leq n$? How many functions $f$ in $F$ satisfy the property $f(i)<f(j)$ for all $i,j \ \ 1\leq i \leq j \leq n$?

Let $F$ be the set of one-to-one functions from the set $\{1, 2, \dots, n\}$ to the set $\{1, 2,\dots, m\}$ where $m\geq n\geq1$.How many functions are members of $F$?How...

Kathleen

6.5k views

Kathleen asked Sep 29, 2014

45 votes

3 answers

98

GATE CSE 2006 | Question: 27
Consider the following propositional statements: $P_1: ((A ∧ B) → C)) ≡ ((A → C) ∧ (B → C))$ $P_2: ((A ∨ B) → C)) ≡ ((A → C) ∨ (B → C))$ Which one of the following is true? $P_1$ is a tautology, but not $P_2$ $P_2$ is a tautology, but not $P_1$ $P_1$ and $P_2$ are both tautologies Both $P_1$ and $P_2$ are not tautologies

Consider the following propositional statements:$P_1: ((A ∧ B) → C)) ≡ ((A → C) ∧ (B → C))$$P_2: ((A ∨ B) → C)) ≡ ((A → C) ∨ (B → C))$Which one of...

Rucha Shelke

8.6k views

Rucha Shelke asked Sep 18, 2014

44 votes

9 answers

99

GATE CSE 2017 Set 2 | Question: 23
$G$ is an undirected graph with $n$ vertices and $25$ edges such that each vertex of $G$ has degree at least $3$. Then the maximum possible value of $n$ is _________ .

$G$ is an undirected graph with $n$ vertices and $25$ edges such that each vertex of $G$ has degree at least $3$. Then the maximum possible value of $n$ is _________ .

Madhav

17.6k views

Madhav asked Feb 14, 2017

44 votes

11 answers

100

GATE CSE 2016 Set 2 | Question: 29
The value of the expression $13^{99}\pmod{17}$ in the range $0$ to $16$, is ________.

The value of the expression $13^{99}\pmod{17}$ in the range $0$ to $16$, is ________.

Akash Kanase

18.0k views

Akash Kanase asked Feb 12, 2016

44 votes

4 answers

101

GATE CSE 2003 | Question: 4
Let $A$ be a sequence of $8$ distinct integers sorted in ascending order. How many distinct pairs of sequences, $B$ and $C$ are there such that each is sorted in ascending order, $B$ has $5$ and $C$ has $3$ elements, and the result of merging $B$ and $C$ gives $A$ $2$ $30$ $56$ $256$

Let $A$ be a sequence of $8$ distinct integers sorted in ascending order. How many distinct pairs of sequences, $B$ and $C$ are there such thateach is sorted in ascending...

Kathleen

13.5k views

Kathleen asked Sep 16, 2014

43 votes

2 answers

102

GATE CSE 1989 | Question: 1-v
The number of possible commutative binary operations that can be defined on a set of $n$ elements (for a given $n$) is ___________.

The number of possible commutative binary operations that can be defined on a set of $n$ elements (for a given $n$) is ___________.

makhdoom ghaya

6.5k views

makhdoom ghaya asked Nov 27, 2016

43 votes

8 answers

103

GATE CSE 2006 | Question: 73
The $2^n$ vertices of a graph $G$ corresponds to all subsets of a set of size $n$, for $n \geq 6$. Two vertices of $G$ are adjacent if and only if the corresponding sets intersect in exactly two elements. The number of connected components in $G$ is: $n$ $n + 2$ $2^{\frac{n}{2}}$ $\frac{2^{n}}{n}$

The $2^n$ vertices of a graph $G$ corresponds to all subsets of a set of size $n$, for $n \geq 6$. Two vertices of $G$ are adjacent if and only if the corresponding set...

go_editor

8.8k views

go_editor asked Apr 24, 2016

43 votes

4 answers

104

GATE CSE 2016 Set 2 | Question: 03
The minimum number of colours that is sufficient to vertex-colour any planar graph is ________.

The minimum number of colours that is sufficient to vertex-colour any planar graph is ________.

Akash Kanase

15.5k views

Akash Kanase asked Feb 12, 2016

43 votes

7 answers

105

GATE CSE 2015 Set 3 | Question: 41
Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p-s=q-r$. Which one of the following is true about $R$? Both reflexive and symmetric Reflexive but not symmetric Not reflexive but symmetric Neither reflexive nor symmetric

Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p-s=q-r$. Which one of the following is true about ...

go_editor

13.0k views

go_editor asked Feb 15, 2015

43 votes

11 answers

106

GATE IT 2004 | Question: 35
In how many ways can we distribute $5$ distinct balls, $B_1, B_2, \ldots, B_5$ in $5$ distinct cells, $C_1, C_2, \ldots, C_5$ such that Ball $B_i$ is not in cell $C_i$, $\forall i= 1,2,\ldots 5$ and each cell contains exactly one ball? $44$ $96$ $120$ $3125$

In how many ways can we distribute $5$ distinct balls, $B_1, B_2, \ldots, B_5$ in $5$ distinct cells, $C_1, C_2, \ldots, C_5$ such that Ball $B_i$ is not in cell $C_i$, $...

Ishrat Jahan

11.5k views

Ishrat Jahan asked Nov 2, 2014

43 votes

5 answers

107

GATE IT 2007 | Question: 21
Which one of these first-order logic formulae is valid? $\forall x\left(P\left(x\right) \implies Q\left(x\right)\right) \implies \left(∀xP\left(x\right)\implies \forall xQ\left(x\right)\right)$ ... $\forall x \exists y P\left(x, y\right)\implies \exists y \forall x P\left(x, y\right)$

Which one of these first-order logic formulae is valid?$\forall x\left(P\left(x\right) \implies Q\left(x\right)\right) \implies \left(∀xP\left(x\right)\implies \forall ...

Ishrat Jahan

10.5k views

Ishrat Jahan asked Oct 29, 2014

43 votes

6 answers

108

GATE IT 2008 | Question: 27
$G$ is a simple undirected graph. Some vertices of $G$ are of odd degree. Add a node $v$ to $G$ and make it adjacent to each odd degree vertex of $G$. The resultant graph is sure to be regular complete Hamiltonian Euler

$G$ is a simple undirected graph. Some vertices of $G$ are of odd degree. Add a node $v$ to $G$ and make it adjacent to each odd degree vertex of $G$. The resultant graph...

Ishrat Jahan

14.1k views

Ishrat Jahan asked Oct 28, 2014

43 votes

2 answers

109

GATE IT 2008 | Question: 22
Which of the following is the negation of $[∀ x, α → (∃y, β → (∀ u, ∃v, y))]$ $[∃ x, α → (∀y, β → (∃u, ∀ v, y))]$ $[∃ x, α → (∀y, β → (∃u, ∀ v, ¬y))]$ $[∀ x, ¬α → (∃y, ¬β → (∀u, ∃ v, ¬y))]$ $[∃ x, α \wedge (∀y, β \wedge (∃u, ∀ v, ¬y))]$

Which of the following is the negation of $[∀ x, α → (∃y, β → (∀ u, ∃v, y))]$$[∃ x, α → (∀y, β → (∃u, ∀ v, y))]$$[∃ x, α → (∀y, β → ...

Ishrat Jahan

7.8k views

Ishrat Jahan asked Oct 27, 2014

43 votes

9 answers

110

GATE CSE 1996 | Question: 2.1
Let $R$ denote the set of real numbers. Let $f:R\times R \rightarrow R \times R$ be a bijective function defined by $f(x,y) = (x+y, x-y)$. The inverse function of $f$ is given by $f^{-1} (x,y) = \left( \frac {1}{x+y}, \frac{1}{x-y}\right)$ ... $f^{-1}(x,y)=\left [ 2\left(x-y\right),2\left(x+y\right) \right ]$

Let $R$ denote the set of real numbers. Let $f:R\times R \rightarrow R \times R$ be a bijective function defined by $f(x,y) = (x+y, x-y)$. The inverse function of $f$ is ...

Kathleen

9.9k views

Kathleen asked Oct 9, 2014

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