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Highest voted questions in Discrete Mathematics
65
votes
16
answers
31
GATE CSE 2015 Set 3 | Question: 5
The number of $4$ digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set $\{1, 2, 3\}$ is ________.
go_editor
asked
in
Combinatory
Feb 14, 2015
by
go_editor
15.5k
views
gatecse-2015-set3
combinatory
normal
numerical-answers
65
votes
4
answers
32
GATE CSE 2006 | Question: 71
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 vertices of degree zero in $G$ is: $1$ $n$ $n + 1$ $2^n$
Rucha Shelke
asked
in
Graph Theory
Sep 26, 2014
by
Rucha Shelke
17.1k
views
gatecse-2006
graph-theory
normal
degree-of-graph
65
votes
9
answers
33
GATE CSE 2004 | Question: 75
Mala has the colouring book in which each English letter is drawn two times. She wants to paint each of these $52$ prints with one of $k$ colours, such that the colour pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of $k$ that satisfies this requirement? $9$ $8$ $7$ $6$
Kathleen
asked
in
Combinatory
Sep 18, 2014
by
Kathleen
16.6k
views
gatecse-2004
combinatory
65
votes
9
answers
34
GATE CSE 2003 | Question: 40
A graph $G=(V,E)$ satisfies $\mid E \mid \leq 3 \mid V \mid - 6$. The min-degree of $G$ is defined as $\min_{v\in V}\left\{ \text{degree }(v)\right \}$. Therefore, min-degree of $G$ cannot be $3$ $4$ $5$ $6$
Kathleen
asked
in
Graph Theory
Sep 17, 2014
by
Kathleen
15.6k
views
gatecse-2003
graph-theory
normal
degree-of-graph
65
votes
5
answers
35
GATE CSE 2003 | Question: 8, ISRO2009-53
Let $\text{G}$ be an arbitrary graph with $n$ nodes and $k$ components. If a vertex is removed from $\text{G}$, the number of components in the resultant graph must necessarily lie down between $k$ and $n$ $k-1$ and $k+1$ $k-1$ and $n-1$ $k+1$ and $n-k$
Kathleen
asked
in
Graph Theory
Sep 16, 2014
by
Kathleen
15.3k
views
gatecse-2003
graph-theory
graph-connectivity
normal
isro2009
64
votes
6
answers
36
GATE CSE 2015 Set 1 | Question: 16
For a set $A$, the power set of $A$ is denoted by $2^{A}$. If $A = \left\{5,\left\{6\right\}, \left\{7\right\}\right\}$, which of the following options are TRUE? $\varnothing \in 2^{A}$ $\varnothing \subseteq 2^{A}$ ... I and III only II and III only I, II and III only I, II and IV only
makhdoom ghaya
asked
in
Set Theory & Algebra
Feb 13, 2015
by
makhdoom ghaya
15.4k
views
gatecse-2015-set1
set-theory&algebra
set-theory
normal
64
votes
9
answers
37
GATE IT 2006 | Question: 25
Consider the undirected graph $G$ defined as follows. The vertices of $G$ are bit strings of length $n$. We have an edge between vertex $u$ and vertex $v$ if and only if $u$ and $v$ differ in exactly one bit position (in other words, $v$ can be obtained from $u$ by ... $\left(\frac{1}{n}\right)$ $\left(\frac{2}{n}\right)$ $\left(\frac{3}{n}\right)$
Ishrat Jahan
asked
in
Graph Theory
Oct 31, 2014
by
Ishrat Jahan
13.1k
views
gateit-2006
graph-theory
graph-coloring
normal
63
votes
5
answers
38
GATE CSE 2014 Set 3 | Question: 2
Let $X$ and $Y$ be finite sets and $f:X \to Y$ be a function. Which one of the following statements is TRUE? For any subsets $A$ and $B$ of $X, |f(A \cup B)| = |f(A)| + |f(B)|$ For any subsets $A$ and $B$ of $X, f(A \cap B) = f(A) \cap f(B)$ For any subsets $A$ ... $S$ and $T$ of $Y, f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$
go_editor
asked
in
Set Theory & Algebra
Sep 28, 2014
by
go_editor
14.0k
views
gatecse-2014-set3
set-theory&algebra
functions
normal
63
votes
14
answers
39
GATE CSE 2014 Set 1 | Question: 49
A pennant is a sequence of numbers, each number being $1$ or $2$. An $n-$pennant is a sequence of numbers with sum equal to $n$. For example, $(1,1,2)$ is a $4-$pennant. The set of all possible $1-$pennants is ${(1)}$, the set of all possible ... $(1,2)$ is not the same as the pennant $(2,1)$. The number of $10-$pennants is________
go_editor
asked
in
Combinatory
Sep 28, 2014
by
go_editor
11.3k
views
gatecse-2014-set1
combinatory
numerical-answers
normal
62
votes
6
answers
40
GATE IT 2006 | Question: 21
Consider the following first order logic formula in which $R$ is a binary relation symbol. $∀x∀y (R(x, y) \implies R(y, x))$ The formula is satisfiable and valid satisfiable and so is its negation unsatisfiable but its negation is valid satisfiable but its negation is unsatisfiable
Ishrat Jahan
asked
in
Mathematical Logic
Oct 31, 2014
by
Ishrat Jahan
13.3k
views
gateit-2006
mathematical-logic
normal
first-order-logic
62
votes
10
answers
41
GATE CSE 2002 | Question: 1.8
"If $X$ then $Y$ unless $Z$" is represented by which of the following formulas in propositional logic? ("$\neg$" is negation, "$\land$" is conjunction, and "$\rightarrow$" is implication) $(X\land \neg Z) \rightarrow Y$ $(X \land Y) \rightarrow \neg Z$ $X \rightarrow(Y\land \neg Z)$ $(X \rightarrow Y)\land \neg Z$
Kathleen
asked
in
Mathematical Logic
Sep 15, 2014
by
Kathleen
14.6k
views
gatecse-2002
mathematical-logic
normal
propositional-logic
60
votes
6
answers
42
GATE CSE 2014 Set 1 | Question: 50
Let ܵ$S$ denote the set of all functions $f:\{0,1\}^4 \to \{0,1\}$. Denote by $N$ the number of functions from S to the set $\{0,1\}$. The value of $ \log_2 \log_2N $ is _______.
go_editor
asked
in
Set Theory & Algebra
Sep 28, 2014
by
go_editor
12.4k
views
gatecse-2014-set1
set-theory&algebra
functions
combinatory
numerical-answers
60
votes
5
answers
43
GATE CSE 2007 | Question: 21
How many different non-isomorphic Abelian groups of order $4$ are there? $2$ $3$ $4$ $5$
Kathleen
asked
in
Set Theory & Algebra
Sep 21, 2014
by
Kathleen
19.5k
views
gatecse-2007
group-theory
normal
60
votes
9
answers
44
GATE CSE 2005 | Question: 44
What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs $(a,b)$ and $(c,d)$ in the chosen set such that, $a \equiv c\mod 3$ and $b \equiv d \mod 5$ $4$ $6$ $16$ $24$
gatecse
asked
in
Combinatory
Sep 21, 2014
by
gatecse
13.3k
views
gatecse-2005
set-theory&algebra
normal
pigeonhole-principle
60
votes
9
answers
45
GATE CSE 2003 | Question: 36
How many perfect matching are there in a complete graph of $6$ vertices? $15$ $24$ $30$ $60$
Kathleen
asked
in
Graph Theory
Sep 16, 2014
by
Kathleen
43.8k
views
gatecse-2003
graph-theory
graph-matching
normal
60
votes
6
answers
46
GATE CSE 2000 | Question: 2.6
Let $P(S)$ denotes the power set of set $S.$ Which of the following is always true? $P(P(S)) = P(S)$ $P(S) ∩ P(P(S)) = \{ Ø \}$ $P(S) ∩ S = P(S)$ $S ∉ P(S)$
Kathleen
asked
in
Set Theory & Algebra
Sep 14, 2014
by
Kathleen
13.4k
views
gatecse-2000
set-theory&algebra
easy
set-theory
59
votes
8
answers
47
GATE CSE 2013 | Question: 26
The line graph $L(G)$ of a simple graph $G$ is defined as follows: There is exactly one vertex $v(e)$ in $L(G)$ for each edge $e$ in $G$. For any two edges $e$ and $e'$ in $G$, $L(G)$ has an edge between $v(e)$ and $v(e')$, if and only if ... planar graph is planar. (S) The line graph of a tree is a tree. $P$ only $P$ and $R$ only $R$ only $P, Q$ and $S$ only
Arjun
asked
in
Graph Theory
Sep 24, 2014
by
Arjun
19.0k
views
gatecse-2013
graph-theory
normal
graph-connectivity
59
votes
6
answers
48
GATE CSE 2003 | Question: 37
Let \(f : A \to B\) be an injective (one-to-one) function. Define \(g : 2^A \to 2^B\) as: \(g(C) = \left \{f(x) \mid x \in C\right\} \), for all subsets $C$ of $A$. Define \(h : 2^B \to 2^A\) as: \(h(D) = \{x \mid x \in A, f(x) \in D\}\), for all ... always true? \(g(h(D)) \subseteq D\) \(g(h(D)) \supseteq D\) \(g(h(D)) \cap D = \phi\) \(g(h(D)) \cap (B - D) \ne \phi\)
Kathleen
asked
in
Set Theory & Algebra
Sep 16, 2014
by
Kathleen
8.1k
views
gatecse-2003
set-theory&algebra
functions
difficult
59
votes
7
answers
49
GATE CSE 2003 | Question: 32
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) $((∀x)[α] ⇒ (∀x)[β]) ⇒ (∀x)[α ⇒ β]$ $(∀x)[α] ⇒ (∃x)[α ∧ β]$ $((∀x)[α ∨ β] ⇒ (∃x)[α]) ⇒ (∀x)[α]$ $(∀x)[α ⇒ β] ⇒ (((∀x)[α]) ⇒ (∀x)[β])$
Kathleen
asked
in
Mathematical Logic
Sep 16, 2014
by
Kathleen
16.7k
views
gatecse-2003
mathematical-logic
first-order-logic
normal
58
votes
9
answers
50
GATE CSE 2017 Set 2 | Question: 47
If the ordinary generating function of a sequence $\left \{a_n\right \}_{n=0}^\infty$ is $\large \frac{1+z}{(1-z)^3}$, then $a_3-a_0$ is equal to ___________ .
Arjun
asked
in
Combinatory
Feb 14, 2017
by
Arjun
17.6k
views
gatecse-2017-set2
combinatory
generating-functions
numerical-answers
normal
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