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Answers by Warrior
25
votes
61
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.
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:$P:$ $R$ ...
14.7k
views
answered
Aug 7, 2017
Set Theory & Algebra
gatecse-2016-set2
set-theory&algebra
relations
normal
+
–
6
votes
62
GATE CSE 2005 | Question: 7
The time complexity of computing the transitive closure of a binary relation on a set of $n$ elements is known to be: $O(n)$ $O(n \log n)$ $O \left( n^{\frac{3}{2}} \right)$ $O\left(n^3\right)$
The time complexity of computing the transitive closure of a binary relation on a set of $n$ elements is known to be:$O(n)$$O(n \log n)$$O \left( n^{\frac{3}{2}} \right)...
24.7k
views
answered
Aug 7, 2017
Set Theory & Algebra
gatecse-2005
set-theory&algebra
normal
relations
+
–
6
votes
63
GATE CSE 1998 | Question: 10a
Prove by induction that the expression for the number of diagonals in a polygon of $n$ sides is $\frac{n(n-3)}{2}$
Prove by induction that the expression for the number of diagonals in a polygon of $n$ sides is $\frac{n(n-3)}{2}$
3.8k
views
answered
Aug 6, 2017
Set Theory & Algebra
gate1998
set-theory&algebra
descriptive
relations
+
–
6
votes
64
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.
State whether the following statements are TRUE or FALSE:The union of two equivalence relations is also an equivalence relation.
5.5k
views
answered
Aug 6, 2017
Set Theory & Algebra
gate1987
set-theory&algebra
relations
true-false
+
–
18
votes
65
GATE CSE 1987 | Question: 9a
How many binary relations are there on a set $A$ with $n$ elements?
How many binary relations are there on a set $A$ with $n$ elements?
5.9k
views
answered
Aug 6, 2017
Set Theory & Algebra
gate1987
set-theory&algebra
relations
descriptive
+
–
1
votes
66
GATE CSE 2005 | Question: 42
Let $R$ and $S$ be any two equivalence relations on a non-empty set $A$. Which one of the following statements is TRUE? $R$ $∪$ $S$, $R$ $∩$ $S$ are both equivalence relations $R$ $∪$ $S$ is an equivalence relation $R$ $∩$ $S$ is an equivalence relation Neither $R$ $∪$ $S$ nor $R$ $∩$ $S$ are equivalence relations
Let $R$ and $S$ be any two equivalence relations on a non-empty set $A$. Which one of the following statements is TRUE?$R$ $∪$ $S$, $R$ $∩$ $S$ are both equivalence r...
9.2k
views
answered
Aug 6, 2017
Set Theory & Algebra
gatecse-2005
set-theory&algebra
normal
relations
+
–
13
votes
67
GATE CSE 2007 | Question: 2
Let $S$ be a set of $n$ elements. The number of ordered pairs in the largest and the smallest equivalence relations on $S$ are: $n$ and $n$ $n^2$ and $n$ $n^2$ and $0$ $n$ and $1$
Let $S$ be a set of $n$ elements. The number of ordered pairs in the largest and the smallest equivalence relations on $S$ are:$n$ and $n$$n^2$ and $n$$n^2$ and $0$$n$ an...
8.6k
views
answered
Aug 6, 2017
Set Theory & Algebra
gatecse-2007
set-theory&algebra
normal
relations
+
–
15
votes
68
GATE CSE 2006 | Question: 4
A relation $R$ is defined on ordered pairs of integers as follows: $(x,y)R(u,v) \text{ if } x<u \text{ and } y>v$ Then $R$ is: Neither a Partial Order nor an Equivalence Relation A Partial Order but not a Total Order A total Order An Equivalence Relation
A relation $R$ is defined on ordered pairs of integers as follows: $$(x,y)R(u,v) \text{ if } x<u \text{ and } y>v$$ Then $R$ is: Neither a Partial Order nor an Equivale...
7.7k
views
answered
Aug 6, 2017
Set Theory & Algebra
gatecse-2006
set-theory&algebra
normal
relations
+
–
1
votes
69
TIFR CSE 2010 | Part A | Question: 18
Let $X$ be a set of size $n$. How many pairs of sets (A, B) are there that satisfy the condition $A\subseteq B \subseteq X$ ? $2^{n+1}$ $2^{2n}$ $3^{n}$ $2^{n} + 1$ $3^{n + 1}$
Let $X$ be a set of size $n$. How many pairs of sets (A, B) are there that satisfy the condition $A\subseteq B \subseteq X$ ?$2^{n+1}$$2^{2n}$$3^{n}$$2^{n} + 1$$3^{n + 1}...
4.7k
views
answered
Jul 30, 2017
Set Theory & Algebra
tifr2010
set-theory
+
–
1
votes
70
TIFR CSE 2010 | Part A | Question: 15
Let $A, B$ be sets. Let $\bar{A}$ denote the complement of set $A$ (with respect to some fixed universe), and $( A - B)$ denote the set of elements in $A$ which are not in $B$. Set $(A - (A - B))$ is equal to: $B$ $A\cap \bar{B}$ $A - B$ $A\cap B$ $\bar{B}$
Let $A, B$ be sets. Let $\bar{A}$ denote the complement of set $A$ (with respect to some fixed universe), and $( A - B)$ denote the set of elements in $A$ which are not i...
1.5k
views
answered
Jul 30, 2017
Set Theory & Algebra
tifr2010
set-theory&algebra
set-theory
+
–
4
votes
71
GATE CSE 2015 Set 2 | Question: 18
The cardinality of the power set of $\{0, 1, 2, \dots , 10\}$ is _______
The cardinality of the power set of $\{0, 1, 2, \dots , 10\}$ is _______
5.2k
views
answered
Jul 30, 2017
Set Theory & Algebra
gatecse-2015-set2
set-theory&algebra
set-theory
easy
numerical-answers
+
–
14
votes
72
GATE CSE 1993 | Question: 8.4
Let A be a finite set of size n. The number of elements in the power set of $A\times A$ is: $2^{2^n}$ $2^{n^2}$ $\left(2^n\right)^2$ $\left(2^2\right)^n$ None of the above
Let A be a finite set of size n. The number of elements in the power set of $A\times A$ is:$2^{2^n}$$2^{n^2}$$\left(2^n\right)^2$$\left(2^2\right)^n$None of the above
6.9k
views
answered
Jul 30, 2017
Set Theory & Algebra
gate1993
set-theory&algebra
easy
set-theory
+
–
–3
votes
73
GATE IT 2004 | Question: 2
In a class of $200$ students, $125$ students have taken Programming Language course, $85$ students have taken Data Structures course, $65$ students have taken Computer Organization course; $50$ students have taken both Programming Language and Data Structures, $35$ ... all the three courses. How many students have not taken any of the three courses? $15$ $20$ $25$ $30$
In a class of $200$ students, $125$ students have taken Programming Language course, $85$ students have taken Data Structures course, $65$ students have taken Computer Or...
5.4k
views
answered
Jul 29, 2017
Set Theory & Algebra
gateit-2004
set-theory&algebra
easy
set-theory
+
–
–2
votes
74
TIFR CSE 2012 | Part A | Question: 8
How many pairs of sets $(A, B)$ are there that satisfy the condition $A, B \subseteq \left\{1, 2,...,5\right\}, A \cap B = \{\}?$ $125$ $127$ $130$ $243$ $257$
How many pairs of sets $(A, B)$ are there that satisfy the condition $A, B \subseteq \left\{1, 2,...,5\right\}, A \cap B = \{\}?$$125$$127$$130$$243$$257$
3.1k
views
answered
Jul 29, 2017
Set Theory & Algebra
tifr2012
set-theory&algebra
set-theory
+
–
–3
votes
75
GATE CSE 1996 | Question: 2.4
Which one of the following is false? The set of all bijective functions on a finite set forms a group under function composition The set $\{1, 2, \dots p-1\}$ forms a group under multiplication mod $p$, where $p$ is a prime number The set of all strings over a finite ... $\langle G, * \rangle$ if and only if for any pair of elements $a, b \in S, a * b^{-1} \in S$
Which one of the following is false?The set of all bijective functions on a finite set forms a group under function compositionThe set $\{1, 2, \dots p-1\}$ forms a group...
9.6k
views
answered
Jul 29, 2017
Set Theory & Algebra
gate1996
set-theory&algebra
normal
set-theory
group-theory
+
–
2
votes
76
GATE CSE 1998 | Question: 2.4
In a room containing $28$ people, there are $18$ people who speak English, $15$ people who speak Hindi and $22$ people who speak Kannada. $9$ persons speak both English and Hindi, $11$ persons speak both Hindi and Kannada whereas $13$ persons speak both Kannada and English. How many speak all three languages? $9$ $8$ $7$ $6$
In a room containing $28$ people, there are $18$ people who speak English, $15$ people who speak Hindi and $22$ people who speak Kannada. $9$ persons speak both English a...
6.8k
views
answered
Jul 29, 2017
Set Theory & Algebra
gate1998
set-theory&algebra
easy
set-theory
+
–
5
votes
77
GATE CSE 1995 | Question: 1.20
The number of elements in the power set $P(S)$ of the set $S=\{\{\emptyset\}, 1, \{2, 3\}\}$ is: $2$ $4$ $8$ None of the above
The number of elements in the power set $P(S)$ of the set $S=\{\{\emptyset\}, 1, \{2, 3\}\}$ is:$2$$4$$8$None of the above
16.3k
views
answered
Jul 29, 2017
Set Theory & Algebra
gate1995
set-theory&algebra
normal
set-theory
+
–
4
votes
78
GATE CSE 1994 | Question: 3.9
Every subset of a countable set is countable. State whether the above statement is true or false with reason.
Every subset of a countable set is countable.State whether the above statement is true or false with reason.
3.1k
views
answered
Jul 29, 2017
Set Theory & Algebra
gate1994
set-theory&algebra
normal
set-theory
countable-uncountable-set
true-false
+
–
–1
votes
79
GATE CSE 2014 Set 3 | Question: 16
Let $\Sigma$ be a finite non-empty alphabet and let $2^{\Sigma^*}$ be the power set of $\Sigma^*$. Which one of the following is TRUE? Both $2^{\Sigma^*}$ and $\Sigma^*$ are countable $2^{\Sigma^*}$ is countable and $\Sigma^*$ is uncountable $2^{\Sigma^*}$ is uncountable and $\Sigma^*$ is countable Both $2^{\Sigma^*}$ and $\Sigma^*$ are uncountable
Let $\Sigma$ be a finite non-empty alphabet and let $2^{\Sigma^*}$ be the power set of $\Sigma^*$. Which one of the following is TRUE? Both $2^{\Sigma^*}$ and $\Sigma^*$ ...
9.7k
views
answered
Jul 29, 2017
Theory of Computation
gatecse-2014-set3
theory-of-computation
normal
countable-uncountable-set
+
–
3
votes
80
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)) = ...
11.3k
views
answered
Jul 28, 2017
Set Theory & Algebra
gatecse-2006
set-theory&algebra
normal
set-theory
+
–
11
votes
81
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 �...
6.7k
views
answered
Jul 28, 2017
Set Theory & Algebra
gatecse-2006
set-theory&algebra
normal
set-theory
+
–
0
votes
82
TIFR CSE 2016 | Part A | Question: 8
Let $A$ and $B$ be finite sets such that $A \subseteq B$. Then, what is the value of the expression: $ \sum \limits_{C:A \subseteq C \subseteq B} (-1)^{\mid C \setminus A \mid,}$ Where $C \setminus A=\{x \in C : x \notin A \}$? Always $0$ Always $1$ $0$ if $A=B$ and $1$ otherwise $1$ if $A=B$ and $0$ otherwise Depends on the size of the universe
Let $A$ and $B$ be finite sets such that $A \subseteq B$. Then, what is the value of the expression:$$ \sum \limits_{C:A \subseteq C \subseteq B} (-1)^{\mid C \setminus A...
2.7k
views
answered
Jul 28, 2017
Set Theory & Algebra
tifr2016
set-theory&algebra
set-theory
+
–
–2
votes
83
TIFR CSE 2011 | Part A | Question: 12
The action for this problem takes place in an island of Knights and Knaves, where Knights always make true statements and Knaves always make false statements and everybody is either a Knight or a Knave. Two friends A and B lives in a house. The census ... a Knave. A is a Knave and B is a Knight. Both are Knaves. Both are Knights. No conclusion can be drawn.
The action for this problem takes place in an island of Knights and Knaves, where Knights always make true statements and Knaves always make false statements and everybod...
2.1k
views
answered
Jul 20, 2017
Mathematical Logic
tifr2011
mathematical-logic
propositional-logic
+
–
5
votes
84
GATE CSE 2015 Set 2 | Question: 3
Consider the following two statements. $S_1$: If a candidate is known to be corrupt, then he will not be elected $S_2$: If a candidate is kind, he will be elected Which one of the following statements follows from $S_1$ and $S_2$ as per sound inference ... If a person is kind, he is not known to be corrupt If a person is not kind, he is not known to be corrupt
Consider the following two statements.$S_1$: If a candidate is known to be corrupt, then he will not be elected$S_2$: If a candidate is kind, he will be electedWhich one ...
8.9k
views
answered
Jul 20, 2017
Mathematical Logic
gatecse-2015-set2
mathematical-logic
normal
logical-reasoning
+
–
–1
votes
85
CMI2013-A-07
Consider the following two statements. There are infinitely many interesting whole numbers. There are finitely many uninteresting whole numbers. Which of the following is true? Statements $1$ and $2$ are equivalent. Statement $1$ implies statement $2$. Statement $2$ implies statement $1$. None of the above.
Consider the following two statements.There are infinitely many interesting whole numbers.There are finitely many uninteresting whole numbers.Which of the following is tr...
2.4k
views
answered
Jul 20, 2017
Mathematical Logic
cmi2013
mathematical-logic
logical-reasoning
+
–
–3
votes
86
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...
16.9k
views
answered
Jul 18, 2017
Mathematical Logic
gatecse-2003
mathematical-logic
first-order-logic
normal
+
–
1
votes
87
TIFR CSE 2012 | Part B | Question: 3
For a person $p$, let $w(p)$, $A(p, y)$, $L(p)$ and $J(p)$ denote that $p$ is a woman, $p$ admires $y$, $p$ is a lawyer and $p$ is a judge respectively. Which of the following is the correct translation in first order logic of ...
For a person $p$, let $w(p)$, $A(p, y)$, $L(p)$ and $J(p)$ denote that $p$ is a woman, $p$ admires $y$, $p$ is a lawyer and $p$ is a judge respectively. Which of the foll...
2.1k
views
answered
Jul 18, 2017
Mathematical Logic
tifr2012
mathematical-logic
first-order-logic
+
–
4
votes
88
GATE CSE 2012 | Question: 13
What is the correct translation of the following statement into mathematical logic? “Some real numbers are rational” $\exists x (\text{real}(x) \lor \text{rational}(x))$ $\forall x (\text{real}(x) \to \text{rational}(x))$ $\exists x (\text{real}(x) \wedge \text{rational}(x))$ $\exists x (\text{rational}(x) \to \text{real}(x))$
What is the correct translation of the following statement into mathematical logic?“Some real numbers are rational”$\exists x (\text{real}(x) \lor \text{rational}(x))...
8.7k
views
answered
Jul 18, 2017
Mathematical Logic
gatecse-2012
mathematical-logic
easy
first-order-logic
+
–
7
votes
89
GATE CSE 2006 | Question: 26
Which one of the first order predicate calculus statements given below correctly expresses the following English statement? Tigers and lions attack if they are hungry or threatened. ...
Which one of the first order predicate calculus statements given below correctly expresses the following English statement? Tigers and lions attack if they are hungry or ...
9.3k
views
answered
Jul 17, 2017
Mathematical Logic
gatecse-2006
mathematical-logic
normal
first-order-logic
+
–
5
votes
90
GATE CSE 1991 | Question: 03,xii
If $F_1$, $F_2$ and $F_3$ are propositional formulae such that $F_1 \land F_2 \rightarrow F_3$ and $F_1 \land F_2 \rightarrow \sim F_3$ are both tautologies, then which of the following is true: Both $F_1$ and $F_2$ are tautologies The conjunction $F_1 \land F_2$ is not satisfiable Neither is tautologous Neither is satisfiable None of the above
If $F_1$, $F_2$ and $F_3$ are propositional formulae such that $F_1 \land F_2 \rightarrow F_3$ and $F_1 \land F_2 \rightarrow \sim F_3$ are both tautologies, then which ...
9.0k
views
answered
Jul 13, 2017
Mathematical Logic
gate1991
mathematical-logic
normal
propositional-logic
multiple-selects
+
–
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