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Previous GATE
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Previous GATE Questions in Discrete Mathematics
22
votes
4
answers
101
GATE CSE 1987 | Question: 2d
State whether the following statements are TRUE or FALSE: The union of two equivalence relations is also an equivalence relation.
makhdoom ghaya
asked
in
Set Theory & Algebra
Nov 9, 2016
by
makhdoom ghaya
5.5k
views
gate1987
set-theory&algebra
relations
true-false
26
votes
3
answers
102
GATE CSE 1992 | Question: 14b
Consider the set of integers $\{1,2,3,4,6,8,12,24\}$ together with the two binary operations LCM (lowest common multiple) and GCD (greatest common divisor). Which of the following algebraic structures does this represent? group ring field lattice
go_editor
asked
in
Set Theory & Algebra
Apr 24, 2016
by
go_editor
4.6k
views
gate1992
set-theory&algebra
partial-order
lattice
normal
7
votes
2
answers
103
GATE CSE 1992 | Question: 15.b
Let $S$ be the set of all integers and let $n > 1$ be a fixed integer. Define for $a,b \in S, a R b$ iff $a-b$ is a multiple of $n$. Show that $R$ is an equivalence relation and find its equivalence classes for $n = 5$.
go_editor
asked
in
Set Theory & Algebra
Apr 24, 2016
by
go_editor
3.3k
views
gate1992
set-theory&algebra
normal
relations
descriptive
43
votes
8
answers
104
GATE CSE 2006 | Question: 73
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 connected components in $G$ is: $n$ $n + 2$ $2^{\frac{n}{2}}$ $\frac{2^{n}}{n}$
go_editor
asked
in
Graph Theory
Apr 24, 2016
by
go_editor
8.7k
views
gatecse-2006
graph-theory
normal
graph-connectivity
89
votes
6
answers
105
GATE CSE 2006 | Question: 72
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 maximum degree of a vertex in $G$ is: $\binom{\frac{n}{2}}{2}.2^{\frac{n}{2}}$ $2^{n-2}$ $2^{n-3}\times 3$ $2^{n-1}$
go_editor
asked
in
Graph Theory
Apr 24, 2016
by
go_editor
17.7k
views
gatecse-2006
graph-theory
normal
degree-of-graph
36
votes
5
answers
106
GATE CSE 2007 | Question: 85
Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at $(i,j)$ then it can move to either $(i + 1, j)$ or $(i,j + 1)$. Suppose that the robot is not allowed to traverse the ... $^{20}\mathrm{C}_{10} - ^{8}\mathrm{C}_{4}\times ^{11}\mathrm{C}_{5}$
go_editor
asked
in
Combinatory
Apr 23, 2016
by
go_editor
9.5k
views
gatecse-2007
combinatory
normal
discrete-mathematics
91
votes
9
answers
107
GATE CSE 2016 Set 1 | Question: 28
A function $f: \Bbb{N^+} \rightarrow \Bbb{N^+}$ , defined on the set of positive integers $\Bbb{N^+}$, satisfies the following properties: $f(n)=f(n/2)$ if $n$ is even $f(n)=f(n+5)$ if $n$ is odd Let $R=\{ i \mid \exists{j} : f(j)=i \}$ be the set of distinct values that $f$ takes. The maximum possible size of $R$ is ___________.
Sandeep Singh
asked
in
Set Theory & Algebra
Feb 12, 2016
by
Sandeep Singh
21.5k
views
gatecse-2016-set1
set-theory&algebra
functions
normal
numerical-answers
66
votes
10
answers
108
GATE CSE 2016 Set 1 | Question: 27
Consider the recurrence relation $a_1 =8 , a_n =6n^2 +2n+a_{n-1}$. Let $a_{99}=K\times 10^4$. The value of $K$ is __________.
Sandeep Singh
asked
in
Combinatory
Feb 12, 2016
by
Sandeep Singh
28.1k
views
gatecse-2016-set1
combinatory
recurrence-relation
normal
numerical-answers
57
votes
17
answers
109
GATE CSE 2016 Set 1 | Question: 26
The coefficient of $x^{12}$ in $\left(x^{3}+x^{4}+x^{5}+x^{6}+\dots \right)^{3}$ is ___________.
Sandeep Singh
asked
in
Combinatory
Feb 12, 2016
by
Sandeep Singh
25.9k
views
gatecse-2016-set1
combinatory
generating-functions
normal
numerical-answers
71
votes
7
answers
110
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 ____________.
Sandeep Singh
asked
in
Mathematical Logic
Feb 12, 2016
by
Sandeep Singh
12.9k
views
gatecse-2016-set1
mathematical-logic
normal
numerical-answers
propositional-logic
45
votes
4
answers
111
GATE CSE 2016 Set 1 | Question: 2
Let $a_n$ be the number of $n$-bit strings that do NOT contain two consecutive $1's$. Which one of the following is the recurrence relation for $a_n$? $a_n = a_{n-1}+ 2a_{n-2}$ $a_n = a_{n-1}+ a_{n-2}$ $a_n = 2a_{n-1}+ a_{n-2}$ $a_n = 2a_{n-1}+ 2a_{n-2}$
Sandeep Singh
asked
in
Combinatory
Feb 12, 2016
by
Sandeep Singh
9.5k
views
gatecse-2016-set1
combinatory
recurrence-relation
easy
69
votes
6
answers
112
GATE CSE 2016 Set 2 | Question: 27
Which one of the following well-formed formulae in predicate calculus is NOT valid ? $(\forall _{x} p(x) \implies \forall _{x} q(x)) \implies (\exists _{x} \neg p(x) \vee \forall _{x} q(x))$ ... $\forall x (p(x) \vee q(x)) \implies (\forall x p(x) \vee \forall x q(x))$
Akash Kanase
asked
in
Mathematical Logic
Feb 12, 2016
by
Akash Kanase
16.8k
views
gatecse-2016-set2
mathematical-logic
first-order-logic
normal
50
votes
8
answers
113
GATE CSE 2016 Set 2 | Question: 26
A binary relation $R$ on $\mathbb{N} \times \mathbb{N}$ is defined as follows: $(a, b) R(c, d)$ if $a \leq c$ or $b \leq d$. Consider the following propositions: $P:$ $R$ is reflexive. $Q:$ $R$ is transitive. Which one of the following statements is TRUE? ... and $Q$ are true. $P$ is true and $Q$ is false. $P$ is false and $Q$ is true. Both $P$ and $Q$ are false.
Akash Kanase
asked
in
Set Theory & Algebra
Feb 12, 2016
by
Akash Kanase
14.6k
views
gatecse-2016-set2
set-theory&algebra
relations
normal
85
votes
8
answers
114
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.
Akash Kanase
asked
in
Set Theory & Algebra
Feb 12, 2016
by
Akash Kanase
16.6k
views
gatecse-2016-set2
set-theory&algebra
difficult
set-theory
44
votes
10
answers
115
GATE CSE 2016 Set 2 | Question: 29
The value of the expression $13^{99}\pmod{17}$ in the range $0$ to $16$, is ________.
Akash Kanase
asked
in
Combinatory
Feb 12, 2016
by
Akash Kanase
17.8k
views
gatecse-2016-set2
modular-arithmetic
normal
numerical-answers
100
votes
11
answers
116
GATE CSE 2016 Set 2 | Question: 01
Consider the following expressions: $false$ $Q$ $true$ $P\vee Q$ $\neg Q\vee P$ The number of expressions given above that are logically implied by $P \wedge (P \Rightarrow Q)$ is ___________.
Akash Kanase
asked
in
Mathematical Logic
Feb 12, 2016
by
Akash Kanase
19.8k
views
gatecse-2016-set2
mathematical-logic
normal
numerical-answers
propositional-logic
43
votes
4
answers
117
GATE CSE 2016 Set 2 | Question: 03
The minimum number of colours that is sufficient to vertex-colour any planar graph is ________.
Akash Kanase
asked
in
Graph Theory
Feb 12, 2016
by
Akash Kanase
15.4k
views
gatecse-2016-set2
graph-theory
graph-coloring
normal
numerical-answers
41
votes
4
answers
118
GATE CSE 1991 | Question: 15,b
Consider the following first order formula: ... Does it have finite models? Is it satisfiable? If so, give a countable model for it.
ibia
asked
in
Mathematical Logic
Nov 16, 2015
by
ibia
6.3k
views
gate1991
mathematical-logic
first-order-logic
descriptive
48
votes
3
answers
119
GATE CSE 1991 | Question: 16-b
Show that all vertices in an undirected finite graph cannot have distinct degrees, if the graph has at least two vertices.
Arjun
asked
in
Graph Theory
Nov 15, 2015
by
Arjun
4.3k
views
gate1991
graph-theory
degree-of-graph
descriptive
proof
43
votes
7
answers
120
GATE CSE 2015 Set 3 | Question: 41
Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p-s=q-r$. Which one of the following is true about $R$? Both reflexive and symmetric Reflexive but not symmetric Not reflexive but symmetric Neither reflexive nor symmetric
go_editor
asked
in
Set Theory & Algebra
Feb 15, 2015
by
go_editor
12.8k
views
gatecse-2015-set3
set-theory&algebra
relations
normal
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