Search results for set-theory&algebra / GATE Overflow for GATE CSE

39 votes

3 answers

41

GATE IT 2004 | Question: 4
Let $R_{1}$ be a relation from $A = \left \{ 1,3,5,7 \right \}$ to $B = \left \{ 2,4,6,8 \right \}$ and $R_{2}$ be another relation from $B$ to $C = \{1, 2, 3, 4\}$ as defined below: An element $x$ in $A$ is related to an element $y$ in $B$ (under $R_{1}$) if $x + y$ is ... $R_{1}R_{2} = \{(3, 2), (3, 4), (5, 1), (5, 3), (7, 1)\} $

Let $R_{1}$ be a relation from $A = \left \{ 1,3,5,7 \right \}$ to $B = \left \{ 2,4,6,8 \right \}$ and $R_{2}$ be another relation from $B$ to $C = \{1, 2, 3, 4\}$ as de...

Ishrat Jahan

10.3k views

Ishrat Jahan asked Nov 1, 2014

23 votes

2 answers

42

GATE CSE 1999 | Question: 1.2
The number of binary relations on a set with $n$ elements is: $n^2$ $2^n$ $2^{n^2}$ None of the above

The number of binary relations on a set with $n$ elements is:$n^2$$2^n$$2^{n^2}$None of the above

Kathleen

11.3k views

Kathleen asked Sep 23, 2014

2 votes

2 answers

43

GATE CSE 2024 | Set 2 | Question: 53
Let $Z_{n}$ be the group of integers $\{0,1,2, \ldots, n-1\}$ with addition modulo $n$ as the group operation. The number of elements in the group $Z_{2} \times Z_{3} \times Z_{4}$ that are their own inverses is ___________.

Let $Z_{n}$ be the group of integers $\{0,1,2, \ldots, n-1\}$ with addition modulo $n$ as the group operation. The number of elements in the group $Z_{2} \times Z_{3} \ti...

Arjun

2.1k views

Arjun asked Feb 16

46 votes

4 answers

44

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...

Kathleen

9.6k views

Kathleen asked Oct 9, 2014

43 votes

5 answers

45

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

54 votes

5 answers

46

GATE CSE 2001 | Question: 2.2
Consider the following statements: $S_1:$ There exists infinite sets $A$, $B$, $C$ such that $A \cap (B \cup C)$ is finite. $S_2:$ There exists two irrational numbers $x$ and y such that $(x+y)$ ... $S_2$? Only $S_1$ is correct Only $S_2$ is correct Both $S_1$ and $S_2$ are correct None of $S_1$ and $S_2$ is correct

Consider the following statements:$S_1:$ There exists infinite sets $A$, $B$, $C$ such that $A \cap (B \cup C)$ is finite.$S_2:$ There exists two irrational numbers $x$ a...

Kathleen

8.9k views

Kathleen asked Sep 14, 2014

41 votes

5 answers

47

GATE CSE 2012 | Question: 37
How many onto (or surjective) functions are there from an $n$-element $(n ≥ 2)$ set to a $2$-element set? $ 2^{n}$ $2^{n} – 1$ $2^{n} – 2$ $2(2^{n} – 2)$

How many onto (or surjective) functions are there from an $n$-element $(n ≥ 2)$ set to a $2$-element set?$ 2^{n}$$2^{n} – 1$$2^{n} – 2$$2(2^{n} – 2)$

gatecse

9.4k views

gatecse asked Sep 26, 2014

10 votes

1 answer

48

GATE CSE 2023 | Question: 41
Let $X$ be a set and $2^{X}$ denote the powerset of $X$. Define a binary operation $\Delta$ on $2^{X}$ as follows: \[ A \Delta B=(A-B) \cup(B-A) \text {. } \] Let $H=\left(2^{X}, \Delta\right)$. Which of the following statements about $H$ is/are correct? ... $A \in 2^{X},$ the inverse of $A$ is the complement of $A$. For every $A \in 2^{X},$ the inverse of $A$ is $A$.

Let $X$ be a set and $2^{X}$ denote the powerset of $X$.Define a binary operation $\Delta$ on $2^{X}$ as follows:\[A \Delta B=(A-B) \cup(B-A) \text {. }\]Let $H=\left(2^{...

admin

5.6k views

admin asked Feb 15, 2023

27 votes

4 answers

49

GATE CSE 2010 | Question: 4
Consider the set $S = \{1, ω, ω^2\}$, where $ω$ and $ω^2$ are cube roots of unity. If $*$ denotes the multiplication operation, the structure $(S, *)$ forms A Group A Ring An integral domain A field

Consider the set $S = \{1, ω, ω^2\}$, where $ω$ and $ω^2$ are cube roots of unity. If $*$ denotes the multiplication operation, the structure $(S, *)$ formsA GroupA R...

gatecse

9.8k views

gatecse asked Sep 21, 2014

34 votes

6 answers

50

GATE CSE 2004 | Question: 24
Consider the binary relation: $S= \left\{\left(x, y\right) \mid y=x+1 \text{ and } x, y \in \left\{0, 1, 2\right\} \right\}$ The reflexive transitive closure is $S$ ... $\left\{\left(x, y\right) \mid y \leq x \text{ and } x, y \in \left\{0, 1, 2\right\} \right\}$

Consider the binary relation:$S= \left\{\left(x, y\right) \mid y=x+1 \text{ and } x, y \in \left\{0, 1, 2\right\} \right\}$The reflexive transitive closure is $S$ is$\lef...

Kathleen

9.9k views

Kathleen asked Sep 18, 2014

41 votes

3 answers

51

GATE CSE 1994 | Question: 1.10
Some group $(G, o)$ is known to be abelian. Then, which one of the following is true for $G$? $g=g^{-1} \text{ for every } g \in G$ $g=g^2 \text{ for every }g \in G$ $(goh)^2 = g^2oh^2 \text{ for every } g, h \in G$ $G$ is of finite order

Some group $(G, o)$ is known to be abelian. Then, which one of the following is true for $G$?$g=g^{-1} \text{ for every } g \in G$$g=g^2 \text{ for every }g \in G$$(goh)...

Kathleen

10.6k views

Kathleen asked Oct 4, 2014

43 votes

7 answers

52

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

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 ...

go_editor

12.9k views

go_editor asked Feb 15, 2015

9 votes

1 answer

53

GO Classes Test Series 2024 | Mock GATE | Test 11 | Question: 38
A binary relation $\mathrm{R}$ over a set $\mathrm{A}$ is called a "GO Relation" if for all $\mathrm{x}, \mathrm{y}, \mathrm{z}$ $\in A$, if $x R y$ and $x R z$, then $y R z$. Which of the following ... is transitive. If $R$ is a GO relation then $R$ is reflexive. If $R$ is an equivalence relation then $R$ is a GO relation.

A binary relation $\mathrm{R}$ over a set $\mathrm{A}$ is called a "GO Relation" if for all $\mathrm{x}, \mathrm{y}, \mathrm{z}$ $\in A$, if $x R y$ and $x R z$, then $y ...

GO Classes

566 views

GO Classes asked Jan 13

4 votes

1 answer

54

GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 39
For sets $A$ and $B$, let $f: A \rightarrow B$ and $g: B \rightarrow A$ be functions such that $f(g(x))=x$ for each $x \in B$. Which among the following statements is/are correct? The function $f$ must be one-to-one. The function $f$ must be onto. The function g must be one-to-one. The function $g$ must be onto.

For sets $A$ and $B$, let $f: A \rightarrow B$ and $g: B \rightarrow A$ be functions such that $f(g(x))=x$ for each $x \in B$. Which among the following statements is/are...

GO Classes

525 views

GO Classes asked Feb 5

43 votes

2 answers

55

GATE CSE 1989 | Question: 1-v
The number of possible commutative binary operations that can be defined on a set of $n$ elements (for a given $n$) is ___________.

The number of possible commutative binary operations that can be defined on a set of $n$ elements (for a given $n$) is ___________.

makhdoom ghaya

6.5k views

makhdoom ghaya asked Nov 27, 2016

27 votes

4 answers

56

GATE CSE 1995 | Question: 2.17
Let $A$ be the set of all non-singular matrices over real number and let $*$ be the matrix multiplication operation. Then $A$ is closed under $*$ but $\langle A, *\rangle$ is not a semigroup. $\langle A, *\rangle$ is a semigroup but not a monoid. $\langle A, * \rangle$ is a monoid but not a group. $\langle A, *\rangle$ is a a group but not an abelian group.

Let $A$ be the set of all non-singular matrices over real number and let $*$ be the matrix multiplication operation. Then$A$ is closed under $*$ but $\langle A, *\rangle$...

Kathleen

9.9k views

Kathleen asked Oct 8, 2014

40 votes

7 answers

57

GATE CSE 2005 | Question: 43
Let $f: B \to C$ and $g: A \to B$ be two functions and let $h = f o g$. Given that $h$ is an onto function which one of the following is TRUE? $f$ and $g$ should both be onto functions $f$ should be onto but $g$ need not to be onto $g$ should be onto but $f$ need not be onto both $f$ and $g$ need not be onto

Let $f: B \to C$ and $g: A \to B$ be two functions and let $h = f o g$. Given that $h$ is an onto function which one of the following is TRUE?$f$ and $g$ should both be o...

gatecse

10.0k views

gatecse asked Sep 21, 2014

29 votes

8 answers

58

GATE CSE 2008 | Question: 2
If $P, Q, R$ are subsets of the universal set U, then $(P\cap Q\cap R) \cup (P^c \cap Q \cap R) \cup Q^c \cup R^c$ is $Q^c \cup R^c$ $P \cup Q^c \cup R^c$ $P^c \cup Q^c \cup R^c$ U

If $P, Q, R$ are subsets of the universal set U, then $$(P\cap Q\cap R) \cup (P^c \cap Q \cap R) \cup Q^c \cup R^c$$ is$Q^c \cup R^c$$P \cup Q^c \cup R^c$$P^c \cup Q^c \c...

Kathleen

9.4k views

Kathleen asked Sep 11, 2014

8 votes

1 answer

59

GO Classes Test Series 2024 | Mock GATE | Test 11 | Question: 12
Let $A-B$ denote $\{x \in A: x \notin B\}$. If $(A-B) \cup B=A$, which of the following must be true? $B$ is empty $A \subseteq B$ $B \subseteq A$ $(B-A) \cup A=B$

Let $A-B$ denote $\{x \in A: x \notin B\}$. If $(A-B) \cup B=A$, which of the following must be true?$B$ is empty$A \subseteq B$$B \subseteq A$$(B-A) \cup A=B$

GO Classes

626 views

GO Classes asked Jan 13

4 votes

1 answer

60

GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 62
As a refresher, if $R$ is an equivalence relation over a set $A$ and $x \in A$, then the equivalence class of $\boldsymbol{x}$ in $\boldsymbol{R}$, denoted $[x]_R,$ is the set $ [x]_R=\{y \in A \mid x R y\} $ Let's now introduce some ... $\mathrm{I}(\mathrm{R})=n / 2$ and $\mathrm{W}(\mathrm{R})=n / 2$

As a refresher, if $R$ is an equivalence relation over a set $A$ and $x \in A$, then the equivalence class of $\boldsymbol{x}$ in $\boldsymbol{R}$, denoted $[x]_R,$ is th...

GO Classes

492 views

GO Classes asked Jan 28

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