Previous GATE Questions in Discrete Mathematics / GATE Overflow for GATE CSE

7 votes

4 answers

1

GATE CSE 2024 | Set 2 | Question: 2
Let $p$ and $q$ be the following propositions: $p$ : Fail grade can be given. $q$ : Student scores more than $50 \%$ marks. Consider the statement: "Fail grade cannot be given when student scores more than $50 \%$ marks." ... above statement in propositional logic? $q \rightarrow \neg p$ $q \rightarrow p$ $p \rightarrow q$ $\neg p \rightarrow q$

Let $p$ and $q$ be the following propositions:$p$ : Fail grade can be given.$q$ : Student scores more than $50 \%$ marks.Consider the statement: "Fail grade c...

Arjun

3.3k views

Arjun asked Feb 16

2 votes

2 answers

2

GATE CSE 2024 | Set 2 | Question: 7
Let $\text{A}$ be the adjacency matrix of a simple undirected graph $\text{G}$. Suppose $\text{A}$ is its own inverse. Which one of the following statements is always TRUE? $\text{G}$ is a cycle $\text{G}$ is a perfect matching $\text{G}$ is a complete graph There is no such graph $\text{G}$

Let $\text{A}$ be the adjacency matrix of a simple undirected graph $\text{G}$. Suppose $\text{A}$ is its own inverse. Which one of the following statements i...

Arjun

2.8k views

Arjun asked Feb 16

2 votes

2 answers

3

GATE CSE 2024 | Set 2 | Question: 24
Let $\text{P}$ be the partial order defined on the set $\{1,2,3,4\}$ as follows \[ P=\{(x, x) \mid x \in\{1,2,3,4\}\} \cup\{(1,2),(3,2),(3,4)\} \] The number of total orders on $\{1,2,3,4\}$ that contain $\text{P}$ is __________.

Let $\text{P}$ be the partial order defined on the set $\{1,2,3,4\}$ as follows\[P=\{(x, x) \mid x \in\{1,2,3,4\}\} \cup\{(1,2),(3,2),(3,4)\}\]The number of total orders ...

Arjun

2.1k views

Arjun asked Feb 16

1 votes

3 answers

4

GATE CSE 2024 | Set 2 | Question: 50
The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. The chromatic number of the following graph is __________.

The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. The chromatic number of the following graph is __________.

Arjun

1.8k views

Arjun asked Feb 16

2 votes

2 answers

5

GATE CSE 2024 | Set 2 | Question: 53
Let $Z_{n}$ be the group of integers $\{0,1,2, \ldots, n-1\}$ with addition modulo $n$ as the group operation. The number of elements in the group $Z_{2} \times Z_{3} \times Z_{4}$ that are their own inverses is ___________.

Let $Z_{n}$ be the group of integers $\{0,1,2, \ldots, n-1\}$ with addition modulo $n$ as the group operation. The number of elements in the group $Z_{2} \times Z_{3} \ti...

Arjun

2.0k views

Arjun asked Feb 16

0 votes

1 answer

6

GATE CSE 2024 | Set 1 | Question: 22
Let $A$ and $B$ be non-empty finite sets such that there exist one-to-one and onto functions $\text{(i)}$ from $A$ to $B$ and $\text{(ii)}$ from $A \times A$ to $A \cup B$. The number of possible values of $\text{|A|}$ is ___________.

Let $A$ and $B$ be non-empty finite sets such that there exist one-to-one and onto functions $\text{(i)}$ from $A$ to $B$ and $\text{(ii)}$ from $A \times A$ to $A \cup B...

Arjun

1.7k views

Arjun asked Feb 16

0 votes

1 answer

7

GATE CSE 2024 | Set 1 | Question: 41
The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. Let $G$ be any graph with $n$ vertices and chromatic number $k$. Which of the following statements is/are always TRUE? $G$ contains a complete subgraph with ... $n/k$ $G$ contains at least $k(k-1) / 2$ edges $G$ contains a vertex of degree at least $k$

The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. Let $G$ be any graph with $n$ vertices and chromatic nu...

Arjun

2.0k views

Arjun asked Feb 16

3 votes

2 answers

8

GATE CSE 2024 | Set 1 | Question: 42
Consider the operators $\diamond$ and $\square$ defined by $a \diamond b=a+2 b, a \square b=a b$, for positive integers. Which of the following statements is/are TRUE? Operator $\diamond$ ... $\square$ obeys the distributive law Operator $\square$ over the operator $\diamond$ obeys the distributive law

Consider the operators $\diamond$ and $\square$ defined by $a \diamond b=a+2 b, a \square b=a b$, for positive integers. Which of the following statements is/are TRUE?Ope...

Arjun

1.8k views

Arjun asked Feb 16

7 votes

4 answers

9

GATE CSE 2023 | Question: 5
The Lucas sequence $L_{n}$ is defined by the recurrence relation: \[ L_{n}=L_{n-1}+L_{n-2}, \quad \text { for } \quad n \geq 3, \] with $L_{1}=1$ and $L_{2}=3$ ... $L_{n}=\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}$

The Lucas sequence $L_{n}$ is defined by the recurrence relation:\[L_{n}=L_{n-1}+L_{n-2}, \quad \text { for } \quad n \geq 3,\]with $L_{1}=1$ and $L_{2}=3$.Which one of t...

admin

8.0k views

admin asked Feb 15, 2023

23 votes

5 answers

10

GATE CSE 2023 | Question: 16
Geetha has a conjecture about integers, which is of the form \[ \forall x(P(x) \Longrightarrow \exists y Q(x, y)), \] where $P$ is a statement about integers, and $Q$ is a statement about pairs of integers. Which of the following (one or more) option(s) would imply ... $\exists y \forall x(P(x) \Longrightarrow Q(x, y))$ $\exists x(P(x) \wedge \exists y Q(x, y))$

Geetha has a conjecture about integers, which is of the form\[\forall x(P(x) \Longrightarrow \exists y Q(x, y)),\]where $P$ is a statement about integers, and $Q$ is a st...

admin

11.4k views

admin asked Feb 15, 2023

14 votes

3 answers

11

GATE CSE 2023 | Question: 38
Let $U=\{1,2, \ldots, n\},$ where $n$ is a large positive integer greater than $1000.$ Let $k$ be a positive integer less than $n$. Let $A, B$ be subsets of $U$ with $|A|=|B|=k$ and $A \cap B=\emptyset$. We say that a permutation of $U$ separates $A$ from $B$ if ... $2\left(\begin{array}{c}n \\ 2 k\end{array}\right)(n-2 k) !(k !)^{2}$

Let $U=\{1,2, \ldots, n\},$ where $n$ is a large positive integer greater than $1000.$ Let $k$ be a positive integer less than $n$. Let $A, B$ be subsets of $U$ with $|A|...

admin

6.6k views

admin asked Feb 15, 2023

14 votes

3 answers

12

GATE CSE 2023 | Question: 39
Let $f: A \rightarrow B$ be an onto (or surjective) function, where $A$ and $B$ are nonempty sets. Define an equivalence relation $\sim$ on the set $A$ as \[ a_{1} \sim a_{2} \text { if } f\left(a_{1}\right)=f\left(a_{2}\right), \] ... is NOT well-defined. $F$ is an onto (or surjective) function. $F$ is a one-to-one (or injective) function. $F$ is a bijective function.

Let $f: A \rightarrow B$ be an onto (or surjective) function, where $A$ and $B$ are nonempty sets. Define an equivalence relation $\sim$ on the set $A$ as\[a_{1} \sim a_{...

admin

5.8k views

admin asked Feb 15, 2023

10 votes

1 answer

13

GATE CSE 2023 | Question: 41
Let $X$ be a set and $2^{X}$ denote the powerset of $X$. Define a binary operation $\Delta$ on $2^{X}$ as follows: \[ A \Delta B=(A-B) \cup(B-A) \text {. } \] Let $H=\left(2^{X}, \Delta\right)$. Which of the following statements about $H$ is/are correct? ... $A \in 2^{X},$ the inverse of $A$ is the complement of $A$. For every $A \in 2^{X},$ the inverse of $A$ is $A$.

Let $X$ be a set and $2^{X}$ denote the powerset of $X$.Define a binary operation $\Delta$ on $2^{X}$ as follows:\[A \Delta B=(A-B) \cup(B-A) \text {. }\]Let $H=\left(2^{...

admin

5.6k views

admin asked Feb 15, 2023

7 votes

3 answers

14

GATE CSE 2023 | Question: 45
Let $G$ be a simple, finite, undirected graph with vertex set $\left\{v_{1}, \ldots, v_{n}\right\}$. Let $\Delta(G)$ denote the maximum degree of $G$ and let $\mathbb{N}=\{1,2, \ldots\}$ denote the set of all possible colors. Color the vertices ... $\Delta(G)$. The number of colors used is equal to the chromatic number of $G$.

Let $G$ be a simple, finite, undirected graph with vertex set $\left\{v_{1}, \ldots, v_{n}\right\}$. Let $\Delta(G)$ denote the maximum degree of $G$ and let $\mathbb{N}=...

admin

8.3k views

admin asked Feb 15, 2023

0 votes

1 answer

15

GATE CSE 2023 | Memory Based Question: 15
The Lucas sequence $L_n$ is defined by the recurrence relation: $L_n=L_{n-1}+L_{n-2}$, for $n \geq 3$ with $L_1=1$ and $L_2=3$. Which one of the options given is TRUE? $L_n=\left(\frac{1+\sqrt{5}}{2}\right)^n+\left(\frac{1-\sqrt{5}}{3}\right)^n$ ... $L_n=\left(\frac{1+\sqrt{5}}{2}\right)^n+\left(\frac{1-\sqrt{5}}{2}\right)^n$

The Lucas sequence $L_n$ is defined by the recurrence relation:$L_n=L_{n-1}+L_{n-2}$, for $n \geq 3$ with $L_1=1$ and $L_2=3$.Which one of the options given is TRUE?$L_n=...

GO Classes

1.5k views

GO Classes asked Feb 5, 2023

1 votes

1 answer

16

GATE CSE 2023 | Memory Based Question: 16
How many permutations of $U$ separate $A$ from $B?$ $2\left(\begin{array}{c}n \\ 2 k\end{array}\right)(n-2 k) !(k!)^2$ $\left(\begin{array}{c}n \\ 2 k\end{array}\right)(n-2 k)!(n!)$ $n!$ $\left(\begin{array}{c}n \\ 2 k\end{array}\right)(n-2 k) !(k !)^2$

How many permutations of $U$ separate $A$ from $B?$$2\left(\begin{array}{c}n \\ 2 k\end{array}\right)(n-2 k) !(k!)^2$$\left(\begin{array}{c}n \\ 2 k\end{array}\right)(n-2...

GO Classes

1.0k views

GO Classes asked Feb 5, 2023

1 votes

2 answers

17

GATE CSE 2023 | Memory Based Question: 17
Let $x$ be a set, $2^x=$ power $2 \mathrm{k}$ set of $\mathrm{X}$. define A binary operation $\Delta$ on $2^x$ as $A \Delta B=(A-B) \cup(B-A)$. Let $H=\left(2^x, \Delta\right)$, then for every $A \in 2^x$; inverse of $A$ ... $\mathrm{H}$ is a group. $\mathrm{H}$ satisfies inverse prop, but not a group for every $A \in 2^x$; the inverse of $A$ is $A$.

Let $x$ be a set, $2^x=$ power $2 \mathrm{k}$ set of $\mathrm{X}$. define A binary operation $\Delta$ on $2^x$ as $A \Delta B=(A-B) \cup(B-A)$. Let $H=\left(2^x, \Delta\r...

GO Classes

885 views

GO Classes asked Feb 5, 2023

14 votes

2 answers

18

GATE CSE 2022 | Question: 17
Which of the following statements is/are $\text{TRUE}$ for a group $\textit{G}?$ If for all $x,y \in \textit{G}, \; (xy)^{2} = x^{2} y^{2},$ then $\textit{G}$ is commutative. If for all $x \in \textit{G}, \; x^{2} = 1,$ then ... $2,$ then $\textit{G}$ is commutative. If $\textit{G}$ is commutative, then a subgroup of $\textit{G}$ need not be commutative.

Which of the following statements is/are $\text{TRUE}$ for a group $\textit{G}?$If for all $x,y \in \textit{G}, \; (xy)^{2} = x^{2} y^{2},$ then $\textit{G}$ is commutati...

Arjun

7.6k views

Arjun asked Feb 15, 2022

19 votes

5 answers

19

GATE CSE 2022 | Question: 20
Consider a simple undirected graph of $10$ vertices. If the graph is disconnected, then the maximum number of edges it can have is _______________ .

Consider a simple undirected graph of $10$ vertices. If the graph is disconnected, then the maximum number of edges it can have is _______________ .

Arjun

8.7k views

Arjun asked Feb 15, 2022

17 votes

3 answers

20

GATE CSE 2022 | Question: 22
The number of arrangements of six identical balls in three identical bins is _____________ .

The number of arrangements of six identical balls in three identical bins is _____________ .

Arjun

10.4k views

Arjun asked Feb 15, 2022

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