Highest voted questions in Discrete Mathematics / GATE Overflow for GATE CSE

66 votes

6 answers

31

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

65 votes

16 answers

32

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 ________.

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

15.6k views

go_editor asked Feb 14, 2015

65 votes

4 answers

33

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: $1$ $n$ $n + 1$ $2^n$

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 set...

Rucha Shelke

17.2k views

Rucha Shelke asked Sep 26, 2014

65 votes

9 answers

34

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? $9$ $8$ $7$ $6$

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 pa...

Kathleen

16.7k views

Kathleen asked Sep 18, 2014

65 votes

9 answers

35

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 $3$ $4$ $5$ $6$

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-d...

Kathleen

15.6k views

Kathleen asked Sep 17, 2014

65 votes

5 answers

36

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 $n-k$

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 neces...

Kathleen

15.4k views

Kathleen asked Sep 16, 2014

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 ... $\left(\frac{1}{n}\right)$ $\left(\frac{2}{n}\right)$ $\left(\frac{3}{n}\right)$

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 ...

Ishrat Jahan

13.2k views

Ishrat Jahan asked Oct 31, 2014

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 $Y, f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$

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...

go_editor

14.1k views

go_editor asked Sep 28, 2014

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________

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. ...

go_editor

11.5k views

go_editor asked Sep 28, 2014

62 votes

7 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

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

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) $(X\land \neg Z) \rightarrow Y$ $(X \land Y) \rightarrow \neg Z$ $X \rightarrow(Y\land \neg Z)$ $(X \rightarrow Y)\land \neg Z$

"If $X$ then $Y$ unless $Z$" is represented by which of the following formulas in propositional logic? ("$\neg$" is negation, "$\land$" is conjunction, and "$\rightarrow$...

Kathleen

14.8k views

Kathleen asked Sep 15, 2014

61 votes

6 answers

42

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? $g(h(D)) \subseteq D$ $g(h(D)) \supseteq D$ $g(h(D)) \cap D = \phi$ $g(h(D)) \cap (B - D) \ne \phi$

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 ...

Kathleen

8.2k views

Kathleen asked Sep 16, 2014

60 votes

6 answers

43

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 _______.

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

12.5k views

go_editor asked Sep 28, 2014

60 votes

5 answers

44

GATE CSE 2007 | Question: 21
How many different non-isomorphic Abelian groups of order $4$ are there? $2$ $3$ $4$ $5$

How many different non-isomorphic Abelian groups of order $4$ are there?$2$$3$$4$$5$

Kathleen

19.6k views

Kathleen asked Sep 21, 2014

60 votes

9 answers

45

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 $b \equiv d \mod 5$ $4$ $6$ $16$ $24$

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 th...

gatecse

13.5k views

gatecse asked Sep 21, 2014

60 votes

10 answers

46

GATE CSE 2003 | Question: 36
How many perfect matching are there in a complete graph of $6$ vertices? $15$ $24$ $30$ $60$

How many perfect matching are there in a complete graph of $6$ vertices?$15$$24$$30$$60$

Kathleen

50.4k views

Kathleen asked Sep 16, 2014

60 votes

6 answers

47

GATE CSE 2000 | Question: 2.6
Let $P(S)$ denotes the power set of set $S.$ Which of the following is always true? $P(P(S)) = P(S)$ $P(S) ∩ P(P(S)) = \{ Ø \}$ $P(S) ∩ S = P(S)$ $S ∉ P(S)$

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

13.5k views

Kathleen asked Sep 14, 2014

59 votes

8 answers

48

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

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 ...

Arjun

19.1k views

Arjun asked Sep 24, 2014

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

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

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 ___________ .

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

17.7k views

Arjun asked Feb 14, 2017

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