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Featured
Most answered questions in Discrete Mathematics
62
votes
7
answers
91
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
13.2k
views
Ishrat Jahan
asked
Oct 31, 2014
Mathematical Logic
gateit-2006
mathematical-logic
normal
first-order-logic
+
–
37
votes
7
answers
92
GATE IT 2006 | Question: 11
If all the edge weights of an undirected graph are positive, then any subset of edges that connects all the vertices and has minimum total weight is a Hamiltonian cycle grid hypercube tree
If all the edge weights of an undirected graph are positive, then any subset of edges that connects all the vertices and has minimum total weight is aHamiltonian cyclegri...
Ishrat Jahan
8.7k
views
Ishrat Jahan
asked
Oct 31, 2014
Graph Theory
gateit-2006
graph-theory
graph-connectivity
normal
+
–
32
votes
7
answers
93
GATE IT 2007 | Question: 23
A partial order $P$ is defined on the set of natural numbers as follows. Here $\frac{x}{y}$ denotes integer division. $(0, 0) \in P.$ $(a, b) \in P$ if and only if $(a \% 10) \leq (b \% 10$) and $(\frac{a}{10},\frac{b}{10})\in P.$ ... $P$? (i) and (iii) (ii) and (iv) (i) and (iv) (iii) and (iv)
A partial order $P$ is defined on the set of natural numbers as follows. Here $\frac{x}{y}$ denotes integer division.$(0, 0) \in P.$$(a, b) \in P$ if and only if $(a \% 1...
Ishrat Jahan
11.2k
views
Ishrat Jahan
asked
Oct 29, 2014
Set Theory & Algebra
gateit-2007
set-theory&algebra
partial-order
normal
+
–
58
votes
7
answers
94
GATE IT 2008 | Question: 4
What is the size of the smallest $\textsf{MIS}$ (Maximal Independent Set) of a chain of nine nodes? $5$ $4$ $3$ $2$
What is the size of the smallest $\textsf{MIS}$ (Maximal Independent Set) of a chain of nine nodes?$5$$4$$3$$2$
Ishrat Jahan
51.3k
views
Ishrat Jahan
asked
Oct 27, 2014
Graph Theory
gateit-2008
normal
graph-connectivity
+
–
38
votes
7
answers
95
GATE CSE 1999 | Question: 2.2
Two girls have picked $10$ roses, $15$ sunflowers and $15$ daffodils. What is the number of ways they can divide the flowers among themselves? $1638$ $2100$ $2640$ None of the above
Two girls have picked $10$ roses, $15$ sunflowers and $15$ daffodils. What is the number of ways they can divide the flowers among themselves?$1638$$2100$$2640$None of th...
Kathleen
12.0k
views
Kathleen
asked
Sep 23, 2014
Combinatory
gate1999
combinatory
normal
+
–
40
votes
7
answers
96
GATE CSE 2005 | Question: 43
Let $f: B \to C$ and $g: A \to B$ be two functions and let $h = f o g$. Given that $h$ is an onto function which one of the following is TRUE? $f$ and $g$ should both be onto functions $f$ should be onto but $g$ need not to be onto $g$ should be onto but $f$ need not be onto both $f$ and $g$ need not be onto
Let $f: B \to C$ and $g: A \to B$ be two functions and let $h = f o g$. Given that $h$ is an onto function which one of the following is TRUE?$f$ and $g$ should both be o...
gatecse
9.9k
views
gatecse
asked
Sep 21, 2014
Set Theory & Algebra
gatecse-2005
set-theory&algebra
functions
normal
+
–
40
votes
7
answers
97
GATE CSE 2010 | Question: 28
The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph? $7, 6, 5, 4, 4, 3, 2, 1$ $6, 6, 6, 6, 3, 3, 2, 2$ $7, 6, 6, 4, 4, 3, 2, 2$ $8, 7, 7, 6, 4, 2, 1, 1$ I and II III and IV IV only II and IV
The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree...
gatecse
18.4k
views
gatecse
asked
Sep 21, 2014
Graph Theory
gatecse-2010
graph-theory
degree-of-graph
+
–
37
votes
7
answers
98
GATE CSE 2004 | Question: 77
The minimum number of colours required to colour the following graph, such that no two adjacent vertices are assigned the same color, is $2$ $3$ $4$ $5$
The minimum number of colours required to colour the following graph, such that no two adjacent vertices are assigned the same color, is$2$$3$$4$$5$
Kathleen
12.6k
views
Kathleen
asked
Sep 18, 2014
Graph Theory
gatecse-2004
graph-theory
graph-coloring
easy
+
–
87
votes
7
answers
99
GATE CSE 2004 | Question: 23, ISRO2007-32
Identify the correct translation into logical notation of the following assertion. Some boys in the class are taller than all the girls Note: $\text{taller} (x, y)$ is true if $x$ is taller than $y$ ... $(\exists x) (\text{boy}(x) \land (\forall y) (\text{girl}(y) \rightarrow \text{taller}(x, y)))$
Identify the correct translation into logical notation of the following assertion.Some boys in the class are taller than all the girlsNote: $\text{taller} (x, y)$ is true...
Kathleen
114k
views
Kathleen
asked
Sep 18, 2014
Mathematical Logic
gatecse-2004
mathematical-logic
easy
isro2007
first-order-logic
+
–
58
votes
7
answers
100
GATE CSE 2006 | Question: 24
Given a set of elements $N = {1, 2, ..., n}$ and two arbitrary subsets $A⊆N$ and $B⊆N$, how many of the n! permutations $\pi$ from $N$ to $N$ satisfy $\min(\pi(A)) = \min(\pi(B))$, where $\min(S)$ is the smallest integer in the set of integers $S$, and $\pi$(S) is the set of ... $n! \frac{|A ∩ B|}{|A ∪ B|}$ $\dfrac{|A ∩ B|^2}{^n \mathrm{C}_{|A ∪ B|}}$
Given a set of elements $N = {1, 2, ..., n}$ and two arbitrary subsets $A⊆N$ and $B⊆N$, how many of the n! permutations $\pi$ from $N$ to $N$ satisfy $\min(\pi(A)) = ...
Rucha Shelke
11.1k
views
Rucha Shelke
asked
Sep 18, 2014
Set Theory & Algebra
gatecse-2006
set-theory&algebra
normal
set-theory
+
–
25
votes
7
answers
101
GATE CSE 2006 | Question: 22
Let $E, F$ and $G$ be finite sets. Let $X = (E ∩ F) - (F ∩ G)$ and $Y = (E - (E ∩ G)) - (E - F)$. Which one of the following is true? $X ⊂ Y$ $X ⊃ Y$ $X = Y$ $X - Y ≠ \emptyset$ and $Y - X ≠ \emptyset$
Let $E, F$ and $G$ be finite sets. Let$X = (E ∩ F) - (F ∩ G)$ and$Y = (E - (E ∩ G)) - (E - F)$.Which one of the following is true?$X ⊂ Y$$X ⊃ Y$$X = Y$$X - Y �...
Rucha Shelke
6.4k
views
Rucha Shelke
asked
Sep 17, 2014
Set Theory & Algebra
gatecse-2006
set-theory&algebra
normal
set-theory
+
–
59
votes
7
answers
102
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
16.6k
views
Kathleen
asked
Sep 16, 2014
Mathematical Logic
gatecse-2003
mathematical-logic
first-order-logic
normal
+
–
54
votes
7
answers
103
GATE CSE 2009 | Question: 3
Which one of the following is TRUE for any simple connected undirected graph with more than $2$ vertices? No two vertices have the same degree. At least two vertices have the same degree. At least three vertices have the same degree. All vertices have the same degree.
Which one of the following is TRUE for any simple connected undirected graph with more than $2$ vertices? No two vertices have the same degree. At least two vertices ...
gatecse
11.3k
views
gatecse
asked
Sep 15, 2014
Graph Theory
gatecse-2009
graph-theory
normal
degree-of-graph
+
–
28
votes
7
answers
104
GATE CSE 2009 | Question: 26
Consider the following well-formed formulae: $\neg \forall x(P(x))$ $\neg \exists x(P(x))$ $\neg \exists x(\neg P(x))$ $\exists x(\neg P(x))$ Which of the above are equivalent? $\text{I}$ and $\text{III}$ $\text{I}$ and $\text{IV}$ $\text{II}$ and $\text{III}$ $\text{II}$ and $\text{IV}$
Consider the following well-formed formulae:$\neg \forall x(P(x))$$\neg \exists x(P(x))$$\neg \exists x(\neg P(x))$$\exists x(\neg P(x))$Which of the above are equivalent...
gatecse
5.5k
views
gatecse
asked
Sep 15, 2014
Mathematical Logic
gatecse-2009
mathematical-logic
normal
first-order-logic
+
–
47
votes
7
answers
105
GATE CSE 2000 | Question: 2.7
Let $a, b, c, d$ be propositions. Assume that the equivalence $a ⇔ ( b \vee \neg b)$ and $b ⇔c$ hold. Then the truth-value of the formula $(a ∧ b) → (a ∧ c) ∨ d$ is always True False Same as the truth-value of $b$ Same as the truth-value of $d$
Let $a, b, c, d$ be propositions. Assume that the equivalence $a ⇔ ( b \vee \neg b)$ and $b ⇔c$ hold. Then the truth-value of the formula $(a ∧ b) → (a ∧ c) ∨...
Kathleen
12.0k
views
Kathleen
asked
Sep 14, 2014
Mathematical Logic
gatecse-2000
mathematical-logic
normal
propositional-logic
+
–
30
votes
7
answers
106
GATE CSE 2008 | Question: 31
$P$ and $Q$ are two propositions. Which of the following logical expressions are equivalent? $P ∨ \neg Q$ $\neg(\neg P ∧ Q)$ $(P ∧ Q) ∨ (P ∧ \neg Q) ∨ (\neg P ∧ \neg Q)$ $(P ∧ Q) ∨ (P ∧ \neg Q) ∨ (\neg P ∧ Q)$ Only I and II Only I, II and III Only I, II and IV All of I, II, III and IV
$P$ and $Q$ are two propositions. Which of the following logical expressions are equivalent?$P ∨ \neg Q$$\neg(\neg P ∧ Q)$$(P ∧ Q) ∨ (P ∧ \neg Q) ∨ (\neg P �...
Kathleen
8.3k
views
Kathleen
asked
Sep 12, 2014
Mathematical Logic
gatecse-2008
normal
mathematical-logic
propositional-logic
+
–
39
votes
7
answers
107
GATE CSE 2008 | Question: 23
Which of the following statements is true for every planar graph on $n$ vertices? The graph is connected The graph is Eulerian The graph has a vertex-cover of size at most $\frac{3n}{4}$ The graph has an independent set of size at least $\frac{n}{3}$
Which of the following statements is true for every planar graph on $n$ vertices?The graph is connectedThe graph is EulerianThe graph has a vertex-cover of size at most $...
Kathleen
56.6k
views
Kathleen
asked
Sep 11, 2014
Graph Theory
gatecse-2008
graph-theory
normal
graph-planarity
+
–
26
votes
6
answers
108
GATE CSE 2022 | Question: 26
Which one of the following is the closed form for the generating function of the sequence $\{ a_{n} \}_{n \geq 0}$ defined below? $ a_{n} = \left\{\begin{matrix} n + 1, & \text{n is odd} & \\ 1, & \text{otherwise} & \end{matrix}\right.$ ... $\frac{2x}{(1-x^{2})^{2}} + \frac{1}{1-x}$ $\frac{x}{(1-x^{2})^{2}} + \frac{1}{1-x}$
Which one of the following is the closed form for the generating function of the sequence $\{ a_{n} \}_{n \geq 0}$ defined below?$$ a_{n} = \left\{\begin{matrix} n + 1, &...
Arjun
9.3k
views
Arjun
asked
Feb 15, 2022
Combinatory
gatecse-2022
combinatory
generating-functions
2-marks
+
–
15
votes
6
answers
109
GATE CSE 2022 | Question: 41
Consider the following recurrence: $\begin{array}{} f(1) & = & 1; \\ f(2n) & = & 2f(n) - 1, & \; \text{for}\; n \geq 1; \\ f(2n+1) & = & 2f(n) + 1, & \; \text{for}\; n \geq 1. \end{array}$ Then, which of the following statements is/are $\text{TRUE}?$ ... $f(2^{n}) = 1$ $f(5 \cdot 2^{n}) = 2^{n+1} + 1$ $f(2^{n} + 1) = 2^{n} + 1$
Consider the following recurrence:$$\begin{array}{} f(1) & = & 1; \\ f(2n) & = & 2f(n) – 1, & \; \text{for}\; n \geq 1; \\ f(2n+1) & = & 2f(n) + 1, & \; \text...
Arjun
7.6k
views
Arjun
asked
Feb 15, 2022
Combinatory
gatecse-2022
combinatory
recurrence-relation
multiple-selects
2-marks
+
–
25
votes
6
answers
110
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
11.8k
views
Arjun
asked
Feb 18, 2021
Combinatory
gatecse-2021-set2
combinatory
counting
numerical-answers
2-marks
+
–
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