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Previous GATE
Featured
Previous GATE Questions in Discrete Mathematics
59
votes
7
answers
281
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
11.4k
views
Rucha Shelke
asked
Sep 18, 2014
Set Theory & Algebra
gatecse-2006
set-theory&algebra
normal
set-theory
+
–
25
votes
8
answers
282
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
6.8k
views
Rucha Shelke
asked
Sep 17, 2014
Set Theory & Algebra
gatecse-2006
set-theory&algebra
normal
set-theory
+
–
59
votes
6
answers
283
GATE CSE 2003 | Question: 72
The following resolution rule is used in logic programming. Derive clause $(P \vee Q)$ from clauses $(P\vee R),(Q \vee ¬R)$ Which of the following statements related to this rule is FALSE? $((P ∨ R)∧(Q ∨ ¬R))⇒(P ∨ Q)$ ... if $(P ∨ R)∧(Q ∨ ¬R)$ is satisfiable $(P ∨ Q)⇒ \text{FALSE}$ if and only if both $P$ and $Q$ are unsatisfiable
The following resolution rule is used in logic programming.Derive clause $(P \vee Q)$ from clauses $(P\vee R),(Q \vee ¬R)$Which of the following statements related to th...
Kathleen
14.5k
views
Kathleen
asked
Sep 17, 2014
Mathematical Logic
gatecse-2003
mathematical-logic
normal
propositional-logic
+
–
65
votes
9
answers
284
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
15.9k
views
Kathleen
asked
Sep 17, 2014
Graph Theory
gatecse-2003
graph-theory
normal
degree-of-graph
+
–
47
votes
6
answers
285
GATE CSE 2003 | Question: 39
Let $\Sigma = \left\{a, b, c, d, e\right\}$ be an alphabet. We define an encoding scheme as follows: $g(a) = 3, g(b) = 5, g(c) = 7, g(d) = 9, g(e) = 11$. Let $p_i$ denote the i-th prime number $\left(p_1 = 2\right)$ ... numbers is the encoding, $h$, of a non-empty sequence of strings? $2^73^75^7$ $2^83^85^8$ $2^93^95^9$ $2^{10}3^{10}5^{10}$
Let $\Sigma = \left\{a, b, c, d, e\right\}$ be an alphabet. We define an encoding scheme as follows:$g(a) = 3, g(b) = 5, g(c) = 7, g(d) = 9, g(e) = 11$.Let $p_i$ denote t...
Kathleen
7.7k
views
Kathleen
asked
Sep 17, 2014
Set Theory & Algebra
gatecse-2003
set-theory&algebra
functions
normal
+
–
42
votes
10
answers
286
GATE CSE 2003 | Question: 38
Consider the set \(\{a, b, c\}\) with binary operators \(+\) and \(*\) defined as follows: ... $(x, y)$ that satisfy the equations) is $0$ $1$ $2$ $3$
Consider the set \(\{a, b, c\}\) with binary operators \(+\) and \(*\) defined as follows:$$\begin{array}{|c|c|c|c|} \hline \textbf{+} & \textbf{a}& \textbf{b} &\textbf{c...
Kathleen
7.2k
views
Kathleen
asked
Sep 17, 2014
Set Theory & Algebra
gatecse-2003
set-theory&algebra
normal
binary-operation
+
–
62
votes
6
answers
287
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
8.3k
views
Kathleen
asked
Sep 16, 2014
Set Theory & Algebra
gatecse-2003
set-theory&algebra
functions
difficult
+
–
61
votes
10
answers
288
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
50.7k
views
Kathleen
asked
Sep 16, 2014
Graph Theory
gatecse-2003
graph-theory
graph-matching
normal
+
–
23
votes
9
answers
289
GATE CSE 2003 | Question: 34
$m$ identical balls are to be placed in $n$ distinct bags. You are given that $m \geq kn$, where $k$ is a natural number $\geq 1$. In how many ways can the balls be placed in the bags if each bag must contain at least $k$ ... $\left( \begin{array}{c} m - kn + n + k - 2 \\ n - k \end{array} \right)$
$m$ identical balls are to be placed in $n$ distinct bags. You are given that $m \geq kn$, where $k$ is a natural number $\geq 1$. In how many ways can the balls be place...
Kathleen
11.4k
views
Kathleen
asked
Sep 16, 2014
Combinatory
gatecse-2003
combinatory
balls-in-bins
normal
+
–
116
votes
6
answers
290
GATE CSE 2003 | Question: 33
Consider the following formula and its two interpretations \(I_1\) and \(I_2\). \(\alpha: (\forall x)\left[P_x \Leftrightarrow (\forall y)\left[Q_{xy} \Leftrightarrow \neg Q_{yy} \right]\right] \Rightarrow (\forall x)\left[\neg P_x\right]\) \(I_1\) : Domain: ... I_1\) does not Neither \(I_1\) nor \(I_2\) satisfies \(\alpha\) Both \(I_1\) and \(I_2\) satisfies \(\alpha\)
Consider the following formula and its two interpretations \(I_1\) and \(I_2\).\(\alpha: (\forall x)\left[P_x \Leftrightarrow (\forall y)\left[Q_{xy} \Leftrightarrow \neg...
Kathleen
16.1k
views
Kathleen
asked
Sep 16, 2014
Mathematical Logic
gatecse-2003
mathematical-logic
difficult
first-order-logic
+
–
59
votes
7
answers
291
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
17.0k
views
Kathleen
asked
Sep 16, 2014
Mathematical Logic
gatecse-2003
mathematical-logic
first-order-logic
normal
+
–
58
votes
6
answers
292
GATE CSE 2003 | Question: 31
Let $(S, \leq)$ be a partial order with two minimal elements a and b, and a maximum element c. Let P: S \(\to\) {True, False} be a predicate defined on S. Suppose that P(a) = True, P(b) = False and P(x) \(\implies\) P(y) for all $x, y \in S$ satisfying $x \leq y$ ... for all x \(\in\) S such that b ≤ x and x ≠ c P(x) = False for all x \(\in\) S such that a ≤ x and b ≤ x
Let $(S, \leq)$ be a partial order with two minimal elements a and b, and a maximum element c. Let P: S \(\to\) {True, False} be a predicate defined on S. Suppose that P(...
Kathleen
12.0k
views
Kathleen
asked
Sep 16, 2014
Set Theory & Algebra
gatecse-2003
set-theory&algebra
partial-order
normal
propositional-logic
+
–
66
votes
5
answers
293
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
15.5k
views
Kathleen
asked
Sep 16, 2014
Graph Theory
gatecse-2003
graph-theory
graph-connectivity
normal
isro2009
+
–
33
votes
2
answers
294
GATE CSE 2003 | Question: 7
Consider the set $\Sigma^*$ of all strings over the alphabet $\Sigma = \{0, 1\}$. $\Sigma^*$ with the concatenation operator for strings does not form a group forms a non-commutative group does not have a right identity element forms a group if the empty string is removed from $\Sigma^*$
Consider the set $\Sigma^*$ of all strings over the alphabet $\Sigma = \{0, 1\}$. $\Sigma^*$ with the concatenation operator for stringsdoes not form a groupforms a non-c...
Kathleen
9.0k
views
Kathleen
asked
Sep 16, 2014
Set Theory & Algebra
gatecse-2003
set-theory&algebra
group-theory
normal
+
–
43
votes
5
answers
295
GATE CSE 2003 | Question: 5
$n$ couples are invited to a party with the condition that every husband should be accompanied by his wife. However, a wife need not be accompanied by her husband. The number of different gatherings possible at the party is \(^{2n}\mathrm{C}_n\times 2^n\) \(3^n\) \(\frac{(2n)!}{2^n}\) \(^{2n}\mathrm{C}_n\)
$n$ couples are invited to a party with the condition that every husband should be accompanied by his wife. However, a wife need not be accompanied by her husband. The nu...
Kathleen
10.6k
views
Kathleen
asked
Sep 16, 2014
Combinatory
gatecse-2003
combinatory
normal
+
–
44
votes
4
answers
296
GATE CSE 2003 | Question: 4
Let $A$ be a sequence of $8$ distinct integers sorted in ascending order. How many distinct pairs of sequences, $B$ and $C$ are there such that each is sorted in ascending order, $B$ has $5$ and $C$ has $3$ elements, and the result of merging $B$ and $C$ gives $A$ $2$ $30$ $56$ $256$
Let $A$ be a sequence of $8$ distinct integers sorted in ascending order. How many distinct pairs of sequences, $B$ and $C$ are there such thateach is sorted in ascending...
Kathleen
13.5k
views
Kathleen
asked
Sep 16, 2014
Combinatory
gatecse-2003
combinatory
normal
+
–
34
votes
5
answers
297
GATE CSE 2006 | Question: 4
A relation $R$ is defined on ordered pairs of integers as follows: $(x,y)R(u,v) \text{ if } x<u \text{ and } y>v$ Then $R$ is: Neither a Partial Order nor an Equivalence Relation A Partial Order but not a Total Order A total Order An Equivalence Relation
A relation $R$ is defined on ordered pairs of integers as follows: $$(x,y)R(u,v) \text{ if } x<u \text{ and } y>v$$ Then $R$ is: Neither a Partial Order nor an Equivale...
Rucha Shelke
7.8k
views
Rucha Shelke
asked
Sep 16, 2014
Set Theory & Algebra
gatecse-2006
set-theory&algebra
normal
relations
+
–
43
votes
5
answers
298
GATE CSE 2006 | Question: 3
The set $\{1,2,3,5,7,8,9\}$ under multiplication modulo $10$ is not a group. Given below are four possible reasons. Which one of them is false? It is not closed $2$ does not have an inverse $3$ does not have an inverse $8$ does not have an inverse
The set $\{1,2,3,5,7,8,9\}$ under multiplication modulo $10$ is not a group. Given below are four possible reasons. Which one of them is false?It is not closed$2$ does no...
Rucha Shelke
10.0k
views
Rucha Shelke
asked
Sep 16, 2014
Set Theory & Algebra
gatecse-2006
set-theory&algebra
group-theory
normal
+
–
36
votes
4
answers
299
GATE CSE 2006 | Question: 2
Let $X,Y,Z$ be sets of sizes $x, y$ and $z$ respectively. Let $W = X \times Y$ and $E$ be the set of all subsets of $W$. The number of functions from $Z$ to $E$ is $z^{2^{xy}}$ $z \times 2^{xy}$ $z^{2^{x+y}}$ $2^{xyz}$
Let $X,Y,Z$ be sets of sizes $x, y$ and $z$ respectively. Let $W = X \times Y$ and $E$ be the set of all subsets of $W$. The number of functions from $Z$ to $E$ is$z^{2^{...
Rucha Shelke
7.2k
views
Rucha Shelke
asked
Sep 16, 2014
Set Theory & Algebra
gatecse-2006
set-theory&algebra
normal
functions
+
–
34
votes
8
answers
300
GATE CSE 2002 | Question: 13
In how many ways can a given positive integer $n \geq 2$ be expressed as the sum of $2$ positive integers (which are not necessarily distinct). For example, for $n=3$, the number of ways is $2$, i.e., $1+2, 2+1$. Give only ... $n \geq k$ be expressed as the sum of $k$ positive integers (which are not necessarily distinct). Give only the answer without explanation.
In how many ways can a given positive integer $n \geq 2$ be expressed as the sum of $2$ positive integers (which are not necessarily distinct). For example, for $n=3$, th...
Kathleen
7.4k
views
Kathleen
asked
Sep 15, 2014
Combinatory
gatecse-2002
combinatory
normal
descriptive
balls-in-bins
+
–
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