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Previous GATE Questions in Discrete Mathematics
13
votes
5
answers
31
GATE CSE 2021 Set 1 | Question: 16
In an undirected connected planar graph $G$, there are eight vertices and five faces. The number of edges in $G$ is _________.
In an undirected connected planar graph $G$, there are eight vertices and five faces. The number of edges in $G$ is _________.
Arjun
8.2k
views
Arjun
asked
Feb 18, 2021
Graph Theory
gatecse-2021-set1
graph-theory
graph-planarity
numerical-answers
easy
1-mark
+
–
38
votes
3
answers
32
GATE CSE 2021 Set 1 | Question: 19
There are $6$ jobs with distinct difficulty levels, and $3$ computers with distinct processing speeds. Each job is assigned to a computer such that: The fastest computer gets the toughest job and the slowest computer gets the easiest job. Every computer gets at least one job. The number of ways in which this can be done is ___________.
There are $6$ jobs with distinct difficulty levels, and $3$ computers with distinct processing speeds. Each job is assigned to a computer such that:The fastest computer g...
Arjun
11.8k
views
Arjun
asked
Feb 18, 2021
Combinatory
gatecse-2021-set1
combinatory
counting
numerical-answers
1-mark
+
–
23
votes
4
answers
33
GATE CSE 2021 Set 1 | Question: 34
Let $G$ be a group of order $6$, and $H$ be a subgroup of $G$ such that $1<|H|<6$. Which one of the following options is correct? Both $G$ and $H$ are always cyclic $G$ may not be cyclic, but $H$ is always cyclic $G$ is always cyclic, but $H$ may not be cyclic Both $G$ and $H$ may not be cyclic
Let $G$ be a group of order $6$, and $H$ be a subgroup of $G$ such that $1<|H|<6$. Which one of the following options is correct?Both $G$ and $H$ are always cyclic$G$ may...
Arjun
8.3k
views
Arjun
asked
Feb 18, 2021
Set Theory & Algebra
gatecse-2021-set1
set-theory&algebra
group-theory
2-marks
+
–
35
votes
6
answers
34
GATE CSE 2021 Set 1 | Question: 36
Let $G=(V, E)$ be an undirected unweighted connected graph. The diameter of $G$ is defined as: $\text{diam}(G)=\displaystyle \max_{u,v\in V} \{\text{the length of shortest path between $u$ and $v$}\}$ Let $M$ be the adjacency matrix of $G$. Define graph $G_2$ ... $\text{diam}(G_2) = \text{diam}(G)$ $\text{diam}(G)< \text{diam}(G_2)\leq 2\; \text{diam}(G)$
Let $G=(V, E)$ be an undirected unweighted connected graph. The diameter of $G$ is defined as:$$\text{diam}(G)=\displaystyle \max_{u,v\in V} \{\text{the length of shortes...
Arjun
10.0k
views
Arjun
asked
Feb 18, 2021
Graph Theory
gatecse-2021-set1
graph-theory
graph-connectivity
2-marks
+
–
27
votes
5
answers
35
GATE CSE 2021 Set 1 | Question: 43
A relation $R$ is said to be circular if $a\text{R}b$ and $b\text{R}c$ together imply $c\text{R}a$. Which of the following options is/are correct? If a relation $S$ is reflexive and symmetric, then $S$ is an equivalence relation ... and circular, then $S$ is an equivalence relation. If a relation $S$ is transitive and circular, then $S$ is an equivalence relation.
A relation $R$ is said to be circular if $a\text{R}b$ and $b\text{R}c$ together imply $c\text{R}a$.Which of the following options is/are correct?If a relation $S$ is refl...
Arjun
8.2k
views
Arjun
asked
Feb 18, 2021
Set Theory & Algebra
gatecse-2021-set1
multiple-selects
set-theory&algebra
relations
2-marks
+
–
15
votes
5
answers
36
GATE CSE 2020 | Question: 17
Let $\mathcal{R}$ be the set of all binary relations on the set $\{1,2,3\}$. Suppose a relation is chosen from $\mathcal{R}$ at random. The probability that the chosen relation is reflexive (round off to $3$ decimal places) is ______.
Let $\mathcal{R}$ be the set of all binary relations on the set $\{1,2,3\}$. Suppose a relation is chosen from $\mathcal{R}$ at random. The probability that the chosen re...
Arjun
9.2k
views
Arjun
asked
Feb 12, 2020
Set Theory & Algebra
gatecse-2020
numerical-answers
probability
relations
1-mark
+
–
15
votes
4
answers
37
GATE CSE 2020 | Question: 18
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.
Arjun
9.4k
views
Arjun
asked
Feb 12, 2020
Set Theory & Algebra
gatecse-2020
numerical-answers
group-theory
easy
1-mark
+
–
42
votes
9
answers
38
GATE CSE 2020 | Question: 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$
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) \equ...
Arjun
17.3k
views
Arjun
asked
Feb 12, 2020
Mathematical Logic
gatecse-2020
first-order-logic
mathematical-logic
2-marks
+
–
28
votes
8
answers
39
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
40
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
13.7k
views
Arjun
asked
Feb 12, 2020
Graph Theory
gatecse-2020
numerical-answers
graph-theory
graph-coloring
2-marks
+
–
19
votes
2
answers
41
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
4.4k
views
Arjun
asked
Jun 6, 2019
Set Theory & Algebra
gate1995
set-theory&algebra
set-theory
numerical-answers
+
–
28
votes
4
answers
42
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
11.6k
views
Arjun
asked
Feb 7, 2019
Combinatory
gatecse-2019
engineering-mathematics
discrete-mathematics
combinatory
1-mark
+
–
38
votes
9
answers
43
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
17.4k
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
44
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
21.3k
views
Arjun
asked
Feb 7, 2019
Graph Theory
gatecse-2019
engineering-mathematics
discrete-mathematics
graph-theory
graph-connectivity
1-mark
+
–
19
votes
18
answers
45
GATE CSE 2019 | Question: 21
The value of $3^{51} \text{ mod } 5$ is _____
The value of $3^{51} \text{ mod } 5$ is _____
Arjun
18.2k
views
Arjun
asked
Feb 7, 2019
Combinatory
gatecse-2019
numerical-answers
combinatory
modular-arithmetic
1-mark
+
–
68
votes
10
answers
46
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
20.2k
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
47
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
20.5k
views
Arjun
asked
Feb 7, 2019
Graph Theory
gatecse-2019
engineering-mathematics
discrete-mathematics
graph-theory
graph-connectivity
2-marks
+
–
13
votes
4
answers
48
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
49
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
22.5k
views
gatecse
asked
Feb 14, 2018
Mathematical Logic
gatecse-2018
mathematical-logic
normal
first-order-logic
2-marks
+
–
52
votes
6
answers
50
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
22.0k
views
gatecse
asked
Feb 14, 2018
Set Theory & Algebra
gatecse-2018
set-theory&algebra
countable-uncountable-set
normal
2-marks
+
–
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