Hot questions in Discrete Mathematics / GATE Overflow for GATE CSE

44 votes

10 answers

31

GATE CSE 2016 Set 2 | Question: 29
The value of the expression $13^{99}\pmod{17}$ in the range $0$ to $16$, is ________.

The value of the expression $13^{99}\pmod{17}$ in the range $0$ to $16$, is ________.

Akash Kanase

17.9k views

Akash Kanase asked Feb 12, 2016

85 votes

8 answers

32

GATE CSE 2016 Set 2 | Question: 28
Consider a set $U$ of $23$ different compounds in a chemistry lab. There is a subset $S$ of $U$ of $9$ compounds, each of which reacts with exactly $3$ compounds of $U$. Consider the following statements: Each compound in U \ S reacts ... \ S reacts with an even number of compounds. Which one of the above statements is ALWAYS TRUE? Only I Only II Only III None.

Consider a set $U$ of $23$ different compounds in a chemistry lab. There is a subset $S$ of $U$ of $9$ compounds, each of which reacts with exactly $3$ compounds of $U$. ...

Akash Kanase

16.6k views

Akash Kanase asked Feb 12, 2016

78 votes

6 answers

33

GATE CSE 1992 | Question: 92,xv
Which of the following predicate calculus statements is/are valid? $(\forall (x)) P(x) \vee (\forall(x))Q(x) \implies (\forall (x)) (P(x) \vee Q(x))$ $(\exists (x)) P(x) \wedge (\exists (x))Q(x) \implies (\exists (x)) (P(x) \wedge Q(x))$ ... $(\exists (x)) (P(x) \vee Q(x)) \implies \sim (\forall (x)) P(x) \vee (\exists (x)) Q(x)$

Which of the following predicate calculus statements is/are valid?$(\forall (x)) P(x) \vee (\forall(x))Q(x) \implies (\forall (x)) (P(x) \vee Q(x))$$(\exists (x)) P(x) \w...

Arjun

16.6k views

Arjun asked Sep 2, 2014

74 votes

8 answers

34

GATE CSE 2016 Set 1 | Question: 1
Let $p, q, r, s$ represents the following propositions. $p:x\in\left\{8, 9, 10, 11, 12\right\}$ $q:$ $x$ is a composite number. $r:$ $x$ is a perfect square. $s:$ $x$ is a prime number. The integer $x\geq2$ which satisfies $\neg\left(\left(p\Rightarrow q\right) \wedge \left(\neg r \vee \neg s\right)\right)$ is ____________.

Let $p, q, r, s$ represents the following propositions.$p:x\in\left\{8, 9, 10, 11, 12\right\}$$q:$ $x$ is a composite number.$r:$ $x$ is a perfect square.$s:$ $x$ is a pr...

Sandeep Singh

13.1k views

Sandeep Singh asked Feb 12, 2016

26 votes

6 answers

35

GATE CSE 2021 Set 2 | Question: 50
Let $S$ be a set of consisting of $10$ elements. The number of tuples of the form $(A,B)$ such that $A$ and $B$ are subsets of $S$, and $A \subseteq B$ is ___________

Let $S$ be a set of consisting of $10$ elements. The number of tuples of the form $(A,B)$ such that $A$ and $B$ are subsets of $S$, and $A \subseteq B$ is ___________

Arjun

12.0k views

Arjun asked Feb 18, 2021

59 votes

7 answers

36

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) $((∀x)[α] ⇒ (∀x)[β]) ⇒ (∀x)[α ⇒ β]$ $(∀x)[α] ⇒ (∃x)[α ∧ β]$ $((∀x)[α ∨ β] ⇒ (∃x)[α]) ⇒ (∀x)[α]$ $(∀x)[α ⇒ β] ⇒ (((∀x)[α]) ⇒ (∀x)[β])$

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...

Kathleen

16.8k views

Kathleen asked Sep 16, 2014

40 votes

6 answers

37

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

78 votes

6 answers

38

GATE CSE 2014 Set 3 | Question: 49
Consider the set of all functions $f:\{0,1, \dots,2014\} \to \{0,1,\dots, 2014\}$ such that $ f\left(f\left(i\right)\right)=i$, for all $0 \leq i \leq 2014$. Consider the following statements: $P$. For each such function it must be the case that for every ... is CORRECT? $P, Q$ and $R$ are true Only $Q$ and $R$ are true Only $P$ and $Q$ are true Only $R$ is true

Consider the set of all functions $f:\{0,1, \dots,2014\} \to \{0,1,\dots, 2014\}$ such that $ f\left(f\left(i\right)\right)=i$, for all $0 \leq i \leq 2014$. Consider th...

go_editor

15.6k views

go_editor asked Sep 28, 2014

28 votes

6 answers

39

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

28 votes

8 answers

40

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

16.5k views

Arjun asked Feb 12, 2020

66 votes

6 answers

41

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

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?$\varnoth...

makhdoom ghaya

15.5k views

makhdoom ghaya asked Feb 13, 2015

42 votes

11 answers

42

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

22.7k views

gatecse asked Feb 14, 2018

68 votes

10 answers

43

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

20.1k views

Arjun asked Feb 7, 2019

77 votes

8 answers

44

GATE CSE 2014 Set 2 | Question: 50
Consider the following relation on subsets of the set $S$ of integers between $1$ and $2014$. For two distinct subsets $U$ and $V$ of $S$ we say $U\:<\:V$ if the minimum element in the symmetric difference of the two sets is in $U$. Consider the ... $S1$ is true and $S2$ is false $S2$ is true and $S1$ is false Neither $S1$ nor $S2$ is true

Consider the following relation on subsets of the set $S$ of integers between $1$ and $2014$. For two distinct subsets $U$ and $V$ of $S$ we say $U\:<\:V$ if the minimum ...

go_editor

15.9k views

go_editor asked Sep 28, 2014

86 votes

8 answers

45

GATE CSE 2004 | Question: 79
How many graphs on $n$ labeled vertices exist which have at least $\frac{(n^2 - 3n)}{ 2}$ edges ? $^{\left(\frac{n^2-n}{2}\right)}C_{\left(\frac{n^2-3n} {2}\right)}$ $^{{\large\sum\limits_{k=0}^{\left (\frac{n^2-3n}{2} \right )}}.\left(n^2-n\right)}C_k$ $^{\left(\frac{n^2-n}{2}\right)}C_n$ $^{{\large\sum\limits_{k=0}^n}.\left(\frac{n^2-n}{2}\right)}C_k$

How many graphs on $n$ labeled vertices exist which have at least $\frac{(n^2 - 3n)}{ 2}$ edges ?$^{\left(\frac{n^2-n}{2}\right)}C_{\left(\frac{n^2-3n} {2}\right)}$$^{{\l...

Kathleen

14.4k views

Kathleen asked Sep 18, 2014

58 votes

7 answers

46

GATE IT 2008 | Question: 4
What is the size of the smallest $\textsf{MIS}$ (Maximal Independent Set) of a chain of nine nodes? $5$ $4$ $3$ $2$

What is the size of the smallest $\textsf{MIS}$ (Maximal Independent Set) of a chain of nine nodes?$5$$4$$3$$2$

Ishrat Jahan

59.0k views

Ishrat Jahan asked Oct 27, 2014

24 votes

6 answers

47

GATE CSE 1995 | Question: 1.20
The number of elements in the power set $P(S)$ of the set $S=\{\{\emptyset\}, 1, \{2, 3\}\}$ is: $2$ $4$ $8$ None of the above

The number of elements in the power set $P(S)$ of the set $S=\{\{\emptyset\}, 1, \{2, 3\}\}$ is:$2$$4$$8$None of the above

Kathleen

16.3k views

Kathleen asked Oct 8, 2014

77 votes

6 answers

48

GATE CSE 2014 Set 3 | Question: 50
There are two elements $x,\:y$ in a group $(G,*)$ such that every element in the group can be written as a product of some number of $x$'s and $y$'s in some order. It is known that $x*x=y*y=x*y*x*y=y*x*y*x=e$ where $e$ is the identity element. The maximum number of elements in such a group is ____.

There are two elements $x,\:y$ in a group $(G,*)$ such that every element in the group can be written as a product of some number of $x$'s and $y$'s in some order. It is ...

go_editor

15.6k views

go_editor asked Sep 28, 2014

63 votes

7 answers

49

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

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 issatisfiable and validsatisfia...

Ishrat Jahan

13.4k views

Ishrat Jahan asked Oct 31, 2014

50 votes

9 answers

50

GATE IT 2005 | Question: 36
Let $P(x)$ and $Q(x)$ ...

Let $P(x)$ and $Q(x)$ be arbitrary predicates. Which of the following statements is always TRUE?$\left(\left(\forall x \left(P\left(x\right) \vee Q\left(x\right)\right)\r...

Ishrat Jahan

14.8k views

Ishrat Jahan asked Nov 3, 2014

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