Recent questions in Engineering Mathematics / GATE Overflow for GATE CSE

61 votes

6 answers

9201

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

8.2k views

Kathleen asked Sep 16, 2014

60 votes

10 answers

9202

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

50.5k views

Kathleen asked Sep 16, 2014

23 votes

9 answers

9203

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

11.3k views

Kathleen asked Sep 16, 2014

115 votes

6 answers

9204

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

16.0k views

Kathleen asked Sep 16, 2014

59 votes

7 answers

9205

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

16.9k views

Kathleen asked Sep 16, 2014

58 votes

6 answers

9206

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

11.8k views

Kathleen asked Sep 16, 2014

65 votes

5 answers

9207

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

15.4k views

Kathleen asked Sep 16, 2014

33 votes

2 answers

9208

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

43 votes

5 answers

9209

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

10.5k views

Kathleen asked Sep 16, 2014

44 votes

4 answers

9210

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

43 votes

10 answers

9211

GATE CSE 2003 | Question: 3
Let $P(E)$ denote the probability of the event $E$. Given $P(A) = 1$, $P(B) =\dfrac{1}{2}$, the values of $P(A\mid B)$ and $P(B\mid A)$ respectively are $\left(\dfrac{1}{4}\right),\left(\dfrac{1}{2}\right)$ $\left(\dfrac{1}{2}\right),\left(\dfrac{1}{4}\right)$ $\left(\dfrac{1}{2}\right),{1}$ ${1},\left(\dfrac{1}{2}\right)$

Let $P(E)$ denote the probability of the event $E$. Given $P(A) = 1$, $P(B) =\dfrac{1}{2}$, the values of $P(A\mid B)$ and $P(B\mid A)$ respectively are$\left(\dfrac{1}{4...

Kathleen

11.8k views

Kathleen asked Sep 16, 2014

33 votes

5 answers

9212

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

7.7k views

Rucha Shelke asked Sep 16, 2014

43 votes

5 answers

9213

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

9.9k views

Rucha Shelke asked Sep 16, 2014

36 votes

4 answers

9214

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

7.1k views

Rucha Shelke asked Sep 16, 2014

34 votes

8 answers

9215

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

7.3k views

Kathleen asked Sep 15, 2014

28 votes

5 answers

9216

GATE CSE 2002 | Question: 5a
Obtain the eigen values of the matrix$A=\begin {bmatrix} 1 & 2 & 34 & 49 \\ 0 & 2 & 43 & 94 \\ 0 & 0 & -2 & 104 \\ 0 & 0 & 0 & -1 \end{bmatrix}$

Obtain the eigen values of the matrix$$A=\begin {bmatrix} 1 & 2 & 34 & 49 \\ 0 & 2 & 43 & 94 \\ 0 & 0 & -2 & 104 \\ 0 & 0 & 0 & -1 \end{bmatrix}$$

Kathleen

4.7k views

Kathleen asked Sep 15, 2014

17 votes

2 answers

9217

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 ($\square(S), \subseteq$)

$S=\{(1,2), (2,1)\}$ 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 t...

Kathleen

3.1k views

Kathleen asked Sep 15, 2014

18 votes

5 answers

9218

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 $N_f$

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 $...

Kathleen

4.2k views

Kathleen asked Sep 15, 2014

29 votes

6 answers

9219

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

The binary relation $S= \phi \text{(empty set)}$ on a set $A = \left \{ 1,2,3 \right \}$ is Neither reflexive nor symmetricSymmetric and reflexiveTransitive and reflexive...

Kathleen

13.0k views

Kathleen asked Sep 15, 2014

22 votes

5 answers

9220

GATE CSE 2002 | Question: 2.16
Four fair coins are tossed simultaneously. The probability that at least one head and one tail turn up is $\frac{1}{16}$ $\frac{1}{8}$ $\frac{7}{8}$ $\frac{15}{16}$

Four fair coins are tossed simultaneously. The probability that at least one head and one tail turn up is$\frac{1}{16}$$\frac{1}{8}$$\frac{7}{8}$$\frac{15}{16}$

Kathleen

10.6k views

Kathleen asked Sep 15, 2014

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