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Previous GATE Questions in Discrete Mathematics
28
votes
8
answers
41
GATE CSE 2020 | Question: 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 ______.
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 ______.
Arjun
16.6k
views
Arjun
asked
Feb 12, 2020
Combinatory
gatecse-2020
numerical-answers
combinatory
2-marks
+
–
28
votes
6
answers
42
GATE CSE 2020 | Question: 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 _______
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...
Arjun
13.8k
views
Arjun
asked
Feb 12, 2020
Graph Theory
gatecse-2020
numerical-answers
graph-theory
graph-coloring
2-marks
+
–
20
votes
2
answers
43
GATE CSE 1995 | Question: 25b
Determine the number of positive integers $(\leq 720)$ which are not divisible by any of $2,3$ or $5.$
Determine the number of positive integers $(\leq 720)$ which are not divisible by any of $2,3$ or $5.$
Arjun
4.5k
views
Arjun
asked
Jun 6, 2019
Set Theory & Algebra
gate1995
set-theory&algebra
set-theory
numerical-answers
+
–
29
votes
4
answers
44
GATE CSE 2019 | Question: 5
Let $U = \{1, 2, \dots , n\}$ Let $A=\{(x, X) \mid x \in X, X \subseteq U \}$. Consider the following two statements on $\mid A \mid$. $\mid A \mid = n2^{n-1}$ $\mid A \mid = \Sigma_{k=1}^{n} k \begin{pmatrix} n \\ k \end{pmatrix}$ Which of the above statements is/are TRUE? Only I Only II Both I and II Neither I nor II
Let $U = \{1, 2, \dots , n\}$ Let $A=\{(x, X) \mid x \in X, X \subseteq U \}$. Consider the following two statements on $\mid A \mid$.$\mid A \mid = n2^{n-1}$$\mid A \mi...
Arjun
11.7k
views
Arjun
asked
Feb 7, 2019
Combinatory
gatecse-2019
engineering-mathematics
discrete-mathematics
combinatory
1-mark
+
–
38
votes
9
answers
45
GATE CSE 2019 | Question: 10
Let $G$ be an arbitrary group. Consider the following relations on $G$: $R_1: \forall a , b \in G, \: a R_1 b \text{ if and only if } \exists g \in G \text{ such that } a = g^{-1}bg$ ... $R_1$ and $R_2$ $R_1$ only $R_2$ only Neither $R_1$ nor $R_2$
Let $G$ be an arbitrary group. Consider the following relations on $G$:$R_1: \forall a , b \in G, \: a R_1 b \text{ if and only if } \exists g \in G \text{ such that } a ...
Arjun
17.5k
views
Arjun
asked
Feb 7, 2019
Set Theory & Algebra
gatecse-2019
engineering-mathematics
discrete-mathematics
set-theory&algebra
group-theory
1-mark
+
–
33
votes
14
answers
46
GATE CSE 2019 | Question: 12
Let $G$ be an undirected complete graph on $n$ vertices, where $n > 2$. Then, the number of different Hamiltonian cycles in $G$ is equal to $n!$ $(n-1)!$ $1$ $\frac{(n-1)!}{2}$
Let $G$ be an undirected complete graph on $n$ vertices, where $n 2$. Then, the number of different Hamiltonian cycles in $G$ is equal to$n!$$(n-1)!$$1$$\frac{(n-1)!}{2}...
Arjun
21.5k
views
Arjun
asked
Feb 7, 2019
Graph Theory
gatecse-2019
engineering-mathematics
discrete-mathematics
graph-theory
graph-connectivity
1-mark
+
–
19
votes
18
answers
47
GATE CSE 2019 | Question: 21
The value of $3^{51} \text{ mod } 5$ is _____
The value of $3^{51} \text{ mod } 5$ is _____
Arjun
18.3k
views
Arjun
asked
Feb 7, 2019
Combinatory
gatecse-2019
numerical-answers
combinatory
modular-arithmetic
1-mark
+
–
69
votes
10
answers
48
GATE CSE 2019 | Question: 35
Consider the first order predicate formula $\varphi$: $\forall x [ ( \forall z \: z | x \Rightarrow (( z=x) \vee (z=1))) \rightarrow \exists w ( w > x) \wedge (\forall z \: z | w \Rightarrow ((w=z) \vee (z=1)))]$ Here $a \mid b$ denotes ... of all integers Which of the above sets satisfy $\varphi$? $S_1$ and $S_2$ $S_1$ and $S_3$ $S_2$ and $S_3$ $S_1, S_2$ and $S_3$
Consider the first order predicate formula $\varphi$:$\forall x [ ( \forall z \: z | x \Rightarrow (( z=x) \vee (z=1))) \rightarrow \exists w ( w x) \wedge (\forall z \:...
Arjun
20.3k
views
Arjun
asked
Feb 7, 2019
Mathematical Logic
gatecse-2019
engineering-mathematics
discrete-mathematics
mathematical-logic
first-order-logic
2-marks
+
–
40
votes
6
answers
49
GATE CSE 2019 | Question: 38
Let $G$ be any connected, weighted, undirected graph. $G$ has a unique minimum spanning tree, if no two edges of $G$ have the same weight. $G$ has a unique minimum spanning tree, if, for every cut of $G$, there is a unique minimum-weight edge crossing the cut. Which of the following statements is/are TRUE? I only II only Both I and II Neither I nor II
Let $G$ be any connected, weighted, undirected graph.$G$ has a unique minimum spanning tree, if no two edges of $G$ have the same weight.$G$ has a unique minimum spanning...
Arjun
20.6k
views
Arjun
asked
Feb 7, 2019
Graph Theory
gatecse-2019
engineering-mathematics
discrete-mathematics
graph-theory
graph-connectivity
2-marks
+
–
13
votes
4
answers
50
GATE CSE 1998 | Question: 10b
Let $R$ be a binary relation on $A = \{a, b, c, d, e, f, g, h\}$ represented by the following two component digraph. Find the smallest integers $m$ and $n$ such that $m < n$ and $R^m = R^n$.
Let $R$ be a binary relation on $A = \{a, b, c, d, e, f, g, h\}$ represented by the following two component digraph. Find the smallest integers $m$ and $n$ such that $m <...
Arjun
4.3k
views
Arjun
asked
Aug 12, 2018
Set Theory & Algebra
gate1998
descriptive
set-theory&algebra
relations
+
–
78
votes
3
answers
51
GATE CSE 2018 | Question: 28
Consider the first-order logic sentence $\varphi \equiv \exists \: s \: \exists \: t \: \exists \: u \: \forall \: v \: \forall \: w \forall \: x \: \forall \: y \: \psi(s, t, u, v, w, x, y)$ ... or equal to $3$ There exists no model of $\varphi$ with universe size of greater than $7$ Every model of $\varphi$ has a universe of size equal to $7$
Consider the first-order logic sentence$$\varphi \equiv \exists \: s \: \exists \: t \: \exists \: u \: \forall \: v \: \forall \: w \forall \: x \: \forall \: y \: \psi(...
gatecse
22.6k
views
gatecse
asked
Feb 14, 2018
Mathematical Logic
gatecse-2018
mathematical-logic
normal
first-order-logic
2-marks
+
–
52
votes
6
answers
52
GATE CSE 2018 | Question: 27
Let $N$ be the set of natural numbers. Consider the following sets, $P:$ Set of Rational numbers (positive and negative) $Q:$ Set of functions from $\{0,1\}$ to $N$ $R:$ Set of functions from $N$ to $\{0, 1\}$ $S:$ Set of finite subsets of $N$ Which of the above sets are countable? $Q$ and $S$ only $P$ and $S$ only $P$ and $R$ only $P, Q$ and $S$ only
Let $N$ be the set of natural numbers. Consider the following sets,$P:$ Set of Rational numbers (positive and negative)$Q:$ Set of functions from $\{0,1\}$ to $N$$R:$ Set...
gatecse
22.1k
views
gatecse
asked
Feb 14, 2018
Set Theory & Algebra
gatecse-2018
set-theory&algebra
countable-uncountable-set
normal
2-marks
+
–
29
votes
4
answers
53
GATE CSE 2018 | Question: 19
Let $G$ be a finite group on $84$ elements. The size of a largest possible proper subgroup of $G$ is _____
Let $G$ be a finite group on $84$ elements. The size of a largest possible proper subgroup of $G$ is _____
gatecse
12.4k
views
gatecse
asked
Feb 14, 2018
Set Theory & Algebra
gatecse-2018
group-theory
numerical-answers
set-theory&algebra
1-mark
+
–
32
votes
5
answers
54
GATE CSE 2018 | Question: 18
The chromatic number of the following graph is _____
The chromatic number of the following graph is _____
gatecse
12.3k
views
gatecse
asked
Feb 14, 2018
Graph Theory
graph-theory
graph-coloring
numerical-answers
gatecse-2018
1-mark
+
–
42
votes
11
answers
55
GATE CSE 2018 | Question: 1
Which one of the following is a closed form expression for the generating function of the sequence $\{a_n\}$, where $a_n = 2n +3 \text{ for all } n=0, 1, 2, \dots$? $\frac{3}{(1-x)^2}$ $\frac{3x}{(1-x)^2}$ $\frac{2-x}{(1-x)^2}$ $\frac{3-x}{(1-x)^2}$
Which one of the following is a closed form expression for the generating function of the sequence $\{a_n\}$, where $a_n = 2n +3 \text{ for all } n=0, 1, 2, \dots$?$\frac...
gatecse
22.9k
views
gatecse
asked
Feb 14, 2018
Combinatory
gatecse-2018
generating-functions
normal
combinatory
1-mark
+
–
2
votes
1
answer
56
GATE CSE 2017 Set 1 | Question: 31
https://gateoverflow.in/118312/gate2017-1-31 In the explanation of how 1st statement is true they have said that $\lambda$12 + $\lambda$22 <=50 How is this statement arrived at?
https://gateoverflow.in/118312/gate2017-1-31In the explanation of how 1st statement is truethey have said that $\lambda$12 + $\lambda$22 <=50How is this statement arrived...
A_i_$_h
770
views
A_i_$_h
asked
Oct 25, 2017
73
votes
8
answers
57
GATE CSE 2017 Set 1 | Question: 02
Consider the first-order logic sentence $F:\forall x(\exists yR(x,y))$. Assuming non-empty logical domains, which of the sentences below are implied by $F$? $\exists y(\exists xR(x,y))$ $\exists y(\forall xR(x,y))$ $\forall y(\exists xR(x,y))$ $¬\exists x(\forall y¬R(x,y))$ IV only I and IV only II only II and III only
Consider the first-order logic sentence $F:\forall x(\exists yR(x,y))$. Assuming non-empty logical domains, which of the sentences below are implied by $F$?$\exists y(\ex...
khushtak
17.4k
views
khushtak
asked
Feb 14, 2017
Mathematical Logic
gatecse-2017-set1
mathematical-logic
first-order-logic
+
–
30
votes
8
answers
58
GATE CSE 2017 Set 1 | Question: 01
The statement $\left ( ¬p \right ) \Rightarrow \left ( ¬q \right )$ is logically equivalent to which of the statements below? $p \Rightarrow q$ $q \Rightarrow p$ $\left ( ¬q \right ) \vee p$ $\left ( ¬p \right ) \vee q$ I only I and IV only II only II and III only
The statement $\left ( ¬p \right ) \Rightarrow \left ( ¬q \right )$ is logically equivalent to which of the statements below?$p \Rightarrow q$$q \Rightarrow p$$\left ( ...
khushtak
9.1k
views
khushtak
asked
Feb 14, 2017
Mathematical Logic
gatecse-2017-set1
mathematical-logic
propositional-logic
easy
+
–
44
votes
9
answers
59
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
17.7k
views
Madhav
asked
Feb 14, 2017
Graph Theory
gatecse-2017-set2
graph-theory
numerical-answers
degree-of-graph
+
–
58
votes
9
answers
60
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 ___________ .
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
17.8k
views
Arjun
asked
Feb 14, 2017
Combinatory
gatecse-2017-set2
combinatory
generating-functions
numerical-answers
normal
+
–
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