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Previous GATE Questions in Discrete Mathematics
44
votes
6
answers
281
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? $2^73^75^7$ $2^83^85^8$ $2^93^95^9$ $2^{10}3^{10}5^{10}$
Kathleen
asked
in
Set Theory & Algebra
Sep 17, 2014
by
Kathleen
7.5k
views
gatecse-2003
set-theory&algebra
functions
normal
40
votes
10
answers
282
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 $0$ $1$ $2$ $3$
Kathleen
asked
in
Set Theory & Algebra
Sep 17, 2014
by
Kathleen
7.0k
views
gatecse-2003
set-theory&algebra
normal
binary-operation
59
votes
6
answers
283
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? \(g(h(D)) \subseteq D\) \(g(h(D)) \supseteq D\) \(g(h(D)) \cap D = \phi\) \(g(h(D)) \cap (B - D) \ne \phi\)
Kathleen
asked
in
Set Theory & Algebra
Sep 16, 2014
by
Kathleen
8.1k
views
gatecse-2003
set-theory&algebra
functions
difficult
60
votes
9
answers
284
GATE CSE 2003 | Question: 36
How many perfect matching are there in a complete graph of $6$ vertices? $15$ $24$ $30$ $60$
Kathleen
asked
in
Graph Theory
Sep 16, 2014
by
Kathleen
43.8k
views
gatecse-2003
graph-theory
graph-matching
normal
23
votes
9
answers
285
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 $k$ ... $\left( \begin{array}{c} m - kn + n + k - 2 \\ n - k \end{array} \right)$
Kathleen
asked
in
Combinatory
Sep 16, 2014
by
Kathleen
11.2k
views
gatecse-2003
combinatory
balls-in-bins
normal
113
votes
6
answers
286
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 \(\alpha\)
Kathleen
asked
in
Mathematical Logic
Sep 16, 2014
by
Kathleen
15.7k
views
gatecse-2003
mathematical-logic
difficult
first-order-logic
59
votes
7
answers
287
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) $((∀x)[α] ⇒ (∀x)[β]) ⇒ (∀x)[α ⇒ β]$ $(∀x)[α] ⇒ (∃x)[α ∧ β]$ $((∀x)[α ∨ β] ⇒ (∃x)[α]) ⇒ (∀x)[α]$ $(∀x)[α ⇒ β] ⇒ (((∀x)[α]) ⇒ (∀x)[β])$
Kathleen
asked
in
Mathematical Logic
Sep 16, 2014
by
Kathleen
16.8k
views
gatecse-2003
mathematical-logic
first-order-logic
normal
57
votes
6
answers
288
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
Kathleen
asked
in
Set Theory & Algebra
Sep 16, 2014
by
Kathleen
11.7k
views
gatecse-2003
set-theory&algebra
partial-order
normal
propositional-logic
65
votes
5
answers
289
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 $n-k$
Kathleen
asked
in
Graph Theory
Sep 16, 2014
by
Kathleen
15.3k
views
gatecse-2003
graph-theory
graph-connectivity
normal
isro2009
33
votes
2
answers
290
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 $\Sigma^*$
Kathleen
asked
in
Set Theory & Algebra
Sep 16, 2014
by
Kathleen
8.8k
views
gatecse-2003
set-theory&algebra
group-theory
normal
43
votes
5
answers
291
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 \(^{2n}\mathrm{C}_n\times 2^n\) \(3^n\) \(\frac{(2n)!}{2^n}\) \(^{2n}\mathrm{C}_n\)
Kathleen
asked
in
Combinatory
Sep 16, 2014
by
Kathleen
10.4k
views
gatecse-2003
combinatory
normal
44
votes
4
answers
292
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 $A$ $2$ $30$ $56$ $256$
Kathleen
asked
in
Combinatory
Sep 16, 2014
by
Kathleen
13.3k
views
gatecse-2003
combinatory
normal
33
votes
5
answers
293
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
Rucha Shelke
asked
in
Set Theory & Algebra
Sep 16, 2014
by
Rucha Shelke
7.6k
views
gatecse-2006
set-theory&algebra
normal
relations
43
votes
5
answers
294
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
Rucha Shelke
asked
in
Set Theory & Algebra
Sep 16, 2014
by
Rucha Shelke
9.8k
views
gatecse-2006
set-theory&algebra
group-theory
normal
36
votes
4
answers
295
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 $z^{2^{xy}}$ $z \times 2^{xy}$ $z^{2^{x+y}}$ $2^{xyz}$
Rucha Shelke
asked
in
Set Theory & Algebra
Sep 16, 2014
by
Rucha Shelke
7.0k
views
gatecse-2006
set-theory&algebra
normal
functions
34
votes
8
answers
296
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.
Kathleen
asked
in
Combinatory
Sep 15, 2014
by
Kathleen
7.2k
views
gatecse-2002
combinatory
normal
descriptive
balls-in-bins
17
votes
2
answers
297
GATE CSE 2002 | Question: 4
$S=\{(1,2), (2,1)\}$ is binary relation on set $A = \{1,2,3\}$. Is it irreflexive? Add the minimum number of ordered pairs to S to make it an equivalence relation. Give the modified $S$. Let $S=\{a,b\}$ ... binary relation '$\subseteq$ (set inclusion)' on $\square(S)$. Draw the Hasse diagram corresponding to the lattice ($\square(S), \subseteq$)
Kathleen
asked
in
Set Theory & Algebra
Sep 15, 2014
by
Kathleen
3.0k
views
gatecse-2002
set-theory&algebra
normal
lattice
descriptive
18
votes
5
answers
298
GATE CSE 2002 | Question: 3
Let $A$ be a set of $n(>0)$ elements. Let $N_r$ be the number of binary relations on $A$ and let $N_f$ be the number of functions from $A$ to $A$ Give the expression for $N_r,$ in terms of $n.$ Give the expression for $N_f,$ terms of $n.$ Which is larger for all possible $n,N_r$ or $N_f$
Kathleen
asked
in
Set Theory & Algebra
Sep 15, 2014
by
Kathleen
4.1k
views
gatecse-2002
set-theory&algebra
normal
descriptive
relations
29
votes
6
answers
299
GATE CSE 2002 | Question: 2.17
The binary relation $S= \phi \text{(empty set)}$ on a set $A = \left \{ 1,2,3 \right \}$ is Neither reflexive nor symmetric Symmetric and reflexive Transitive and reflexive Transitive and symmetric
Kathleen
asked
in
Set Theory & Algebra
Sep 15, 2014
by
Kathleen
12.9k
views
gatecse-2002
set-theory&algebra
normal
relations
39
votes
8
answers
300
GATE CSE 2002 | Question: 1.25, ISRO2008-30, ISRO2016-6
The maximum number of edges in a $n$-node undirected graph without self loops is $n^2$ $\frac{n(n-1)}{2}$ $n-1$ $\frac{(n+1)(n)}{2}$
Kathleen
asked
in
Graph Theory
Sep 15, 2014
by
Kathleen
17.5k
views
gatecse-2002
graph-theory
easy
isro2008
isro2016
graph-connectivity
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