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Highest voted questions in Discrete Mathematics
45
votes
3
answers
91
GATE CSE 2013 | Question: 47
Which one of the following is NOT logically equivalent to $¬∃x(∀ y (α)∧∀z(β ))$ ? $∀ x(∃ z(¬β )→∀ y(α))$ $∀x(∀ z(β )→∃ y(¬α))$ $∀x(∀ y(α)→∃z(¬β ))$ $∀x(∃ y(¬α)→∃z(¬β ))$
gatecse
asked
in
Mathematical Logic
Aug 21, 2014
by
gatecse
11.8k
views
mathematical-logic
normal
marks-to-all
gatecse-2013
first-order-logic
44
votes
9
answers
92
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 _________ .
Madhav
asked
in
Graph Theory
Feb 14, 2017
by
Madhav
17.3k
views
gatecse-2017-set2
graph-theory
numerical-answers
degree-of-graph
44
votes
10
answers
93
GATE CSE 2016 Set 2 | Question: 29
The value of the expression $13^{99}\pmod{17}$ in the range $0$ to $16$, is ________.
Akash Kanase
asked
in
Combinatory
Feb 12, 2016
by
Akash Kanase
17.7k
views
gatecse-2016-set2
modular-arithmetic
normal
numerical-answers
44
votes
5
answers
94
GATE CSE 2015 Set 1 | Question: 26
$\sum\limits_{x=1}^{99}\frac{1}{x(x+1)}$ = ______.
makhdoom ghaya
asked
in
Combinatory
Feb 13, 2015
by
makhdoom ghaya
8.1k
views
gatecse-2015-set1
combinatory
normal
numerical-answers
summation
44
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 $C?$ $n$ $n+1$ $2^{n-1} + 1$ $n!$
Ishrat Jahan
asked
in
Set Theory & Algebra
Nov 3, 2014
by
Ishrat Jahan
11.8k
views
gateit-2005
set-theory&algebra
normal
set-theory
44
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
Ishrat Jahan
asked
in
Set Theory & Algebra
Oct 30, 2014
by
Ishrat Jahan
9.6k
views
gateit-2006
set-theory&algebra
easy
binary-operation
44
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 $i,j \ \ 1\leq i \leq j \leq n$?
Kathleen
asked
in
Set Theory & Algebra
Sep 29, 2014
by
Kathleen
6.4k
views
gate1997
set-theory&algebra
functions
normal
descriptive
44
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
Rucha Shelke
asked
in
Mathematical Logic
Sep 18, 2014
by
Rucha Shelke
8.4k
views
gatecse-2006
mathematical-logic
normal
propositional-logic
44
votes
6
answers
99
GATE CSE 2003 | Question: 39
Let $\Sigma = \left\{a, b, c, d, e\right\}$ be an alphabet. We define an encoding scheme as follows: $g(a) = 3, g(b) = 5, g(c) = 7, g(d) = 9, g(e) = 11$. Let $p_i$ denote the i-th prime number $\left(p_1 = 2\right)$ ... numbers is the encoding, $h$, of a non-empty sequence of strings? $2^73^75^7$ $2^83^85^8$ $2^93^95^9$ $2^{10}3^{10}5^{10}$
Kathleen
asked
in
Set Theory & Algebra
Sep 17, 2014
by
Kathleen
7.5k
views
gatecse-2003
set-theory&algebra
functions
normal
44
votes
4
answers
100
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 $A$ $2$ $30$ $56$ $256$
Kathleen
asked
in
Combinatory
Sep 16, 2014
by
Kathleen
13.3k
views
gatecse-2003
combinatory
normal
43
votes
2
answers
101
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 ___________.
makhdoom ghaya
asked
in
Set Theory & Algebra
Nov 27, 2016
by
makhdoom ghaya
6.5k
views
gate1989
descriptive
set-theory&algebra
binary-operation
43
votes
8
answers
102
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: $n$ $n + 2$ $2^{\frac{n}{2}}$ $\frac{2^{n}}{n}$
go_editor
asked
in
Graph Theory
Apr 24, 2016
by
go_editor
8.6k
views
gatecse-2006
graph-theory
normal
graph-connectivity
43
votes
4
answers
103
GATE CSE 2016 Set 2 | Question: 03
The minimum number of colours that is sufficient to vertex-colour any planar graph is ________.
Akash Kanase
asked
in
Graph Theory
Feb 12, 2016
by
Akash Kanase
15.4k
views
gatecse-2016-set2
graph-theory
graph-coloring
normal
numerical-answers
43
votes
7
answers
104
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
go_editor
asked
in
Set Theory & Algebra
Feb 15, 2015
by
go_editor
12.8k
views
gatecse-2015-set3
set-theory&algebra
relations
normal
43
votes
11
answers
105
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? $44$ $96$ $120$ $3125$
Ishrat Jahan
asked
in
Combinatory
Nov 2, 2014
by
Ishrat Jahan
11.4k
views
gateit-2004
combinatory
normal
balls-in-bins
43
votes
6
answers
106
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
Ishrat Jahan
asked
in
Graph Theory
Oct 28, 2014
by
Ishrat Jahan
13.9k
views
gateit-2008
graph-theory
graph-connectivity
normal
43
votes
2
answers
107
GATE IT 2008 | Question: 22
Which of the following is the negation of $[∀ x, α → (∃y, β → (∀ u, ∃v, y))]$ $[∃ x, α → (∀y, β → (∃u, ∀ v, y))]$ $[∃ x, α → (∀y, β → (∃u, ∀ v, ¬y))]$ $[∀ x, ¬α → (∃y, ¬β → (∀u, ∃ v, ¬y))]$ $[∃ x, α \wedge (∀y, β \wedge (∃u, ∀ v, ¬y))]$
Ishrat Jahan
asked
in
Mathematical Logic
Oct 27, 2014
by
Ishrat Jahan
7.7k
views
gateit-2008
mathematical-logic
normal
first-order-logic
43
votes
8
answers
108
GATE CSE 2007 | Question: 22
Let $\text{ Graph}(x)$ be a predicate which denotes that $x$ is a graph. Let $\text{ Connected}(x)$ be a predicate which denotes that $x$ ... $\forall x \, \Bigl ( \text{ Graph}(x) \implies \lnot \text{ Connected}(x) \Bigr )$
Kathleen
asked
in
Mathematical Logic
Sep 21, 2014
by
Kathleen
8.8k
views
gatecse-2007
mathematical-logic
easy
first-order-logic
43
votes
5
answers
109
GATE CSE 2003 | Question: 5
$n$ couples are invited to a party with the condition that every husband should be accompanied by his wife. However, a wife need not be accompanied by her husband. The number of different gatherings possible at the party is \(^{2n}\mathrm{C}_n\times 2^n\) \(3^n\) \(\frac{(2n)!}{2^n}\) \(^{2n}\mathrm{C}_n\)
Kathleen
asked
in
Combinatory
Sep 16, 2014
by
Kathleen
10.4k
views
gatecse-2003
combinatory
normal
43
votes
5
answers
110
GATE CSE 2006 | Question: 3
The set $\{1,2,3,5,7,8,9\}$ under multiplication modulo $10$ is not a group. Given below are four possible reasons. Which one of them is false? It is not closed $2$ does not have an inverse $3$ does not have an inverse $8$ does not have an inverse
Rucha Shelke
asked
in
Set Theory & Algebra
Sep 16, 2014
by
Rucha Shelke
9.8k
views
gatecse-2006
set-theory&algebra
group-theory
normal
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