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Previous GATE
+59
votes
10
answers
1
GATE2015139
Consider the operations $\textit{f (X, Y, Z) = X'YZ + XY' + Y'Z'}$ and $\textit{g (X, Y, Z) = X'YZ + X'YZ' + XY}$ Which one of the following is correct? Both $\left\{\textit{f} \right\}$ and $\left\{ \textit{g}\right\}$ are ... Only $\left\{ \textit{g}\right\}$ is functionally complete Neither $\left\{ \textit{f}\right\}$ nor $\left\{\textit{g}\right\}$ is functionally complete
asked
Feb 13, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
30.8k
points)

8.4k
views
gate20151
settheory&algebra
functions
difficult
+58
votes
7
answers
2
GATE2016128
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 ___________.
asked
Feb 12, 2016
in
Set Theory & Algebra
by
Sandeep Singh
Loyal
(
7.2k
points)

6.8k
views
gate20161
settheory&algebra
functions
normal
numericalanswers
+55
votes
8
answers
3
GATE2016228
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 with an odd number ... in U \ S reacts with an even number of compounds. Which one of the above statements is ALWAYS TRUE? Only I Only II Only III None.
asked
Feb 12, 2016
in
Set Theory & Algebra
by
Akash Kanase
Boss
(
41.9k
points)

5.6k
views
gate20162
settheory&algebra
difficult
sets
+50
votes
3
answers
4
GATE2014349
Consider the set of all functions $f:\{0,1, \dots,2014\} \to \{0,1,\dots, 2014\}$ such that $ f\left(f\left(i\right)\right)=i$, for all $0 \leq i \leq 2014$. Consider the following statements: $P$ ... the following is CORRECT? $P, Q$ and $R$ are true Only $Q$ and $R$ are true Only $P$ and $Q$ are true Only $R$ is true
asked
Sep 28, 2014
in
Set Theory & Algebra
by
jothee
Veteran
(
106k
points)

5.2k
views
gate20143
settheory&algebra
functions
normal
+50
votes
5
answers
5
GATE2014250
Consider the following relation on subsets of the set $S$ of integers between 1 and 2014. For two distinct subsets $U$ and $V$ of $S$ we say $U\:<\:V$ if the minimum element in the symmetric difference of the two sets is in $U$. Consider the following two statements: $S1$ ... $S2$ are true $S1$ is true and $S2$ is false $S2$ is true and $S1$ is false Neither $S1$ nor $S2$ is true
asked
Sep 28, 2014
in
Set Theory & Algebra
by
jothee
Veteran
(
106k
points)

5k
views
gate20142
settheory&algebra
normal
sets
+50
votes
6
answers
6
GATE200625
Let $S = \{1, 2, 3,\ldots, m\}, m >3.$ Let $X_1,\ldots,X_n$ be subsets of $S$ each of size $3.$ Define a function $f$ from $S$ to the set of natural numbers as, $f(i)$ is the number of sets $X_j$ that contain the element $i.$ That is $f(i)=\left  \left\{j \mid i\in X_j \right\} \right$ then $ \sum_{i=1}^{m} f(i)$ is: $3m$ $3n$ $2m+1$ $2n+1$
asked
Sep 18, 2014
in
Set Theory & Algebra
by
Rucha Shelke
Active
(
3.3k
points)

3k
views
gate2006
settheory&algebra
normal
functions
+44
votes
4
answers
7
GATE2014350
There are two elements $x,\:y$ in a group $(G,*)$ such that every element in the group can be written as a product of some number of $x$'s and $y$'s in some order. It is known that $x*x=y*y=x*y*x*y=y*x*y*x=e$ where $e$ is the identity element. The maximum number of elements in such a group is ____.
asked
Sep 28, 2014
in
Set Theory & Algebra
by
jothee
Veteran
(
106k
points)

5.3k
views
gate20143
settheory&algebra
grouptheory
numericalanswers
normal
+43
votes
5
answers
8
GATE2014150
Let ܵ$S$ denote the set of all functions $f:\{0,1\}^4 \to \{0,1\}$. Denote by $N$ the number of functions from S to the set $\{0,1\}$. The value of $ \log_2 \log_2N $ is _______.
asked
Sep 28, 2014
in
Set Theory & Algebra
by
jothee
Veteran
(
106k
points)

4.5k
views
gate20141
settheory&algebra
functions
permutationandcombination
numericalanswers
+41
votes
6
answers
9
GATE2015134
Suppose $L = \left\{ p, q, r, s, t\right\}$ is a lattice represented by the following Hasse diagram: For any $x, y \in L$, not necessarily distinct , $x \vee y$ and $x \wedge y$ are join and meet of $x, y$ ... $p_r = 0$ $p_r = 1$ $0 < p_r ≤ \frac{1}{5}$ $\frac{1}{5} < p_r < 1$
asked
Feb 13, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
30.8k
points)

5.9k
views
gate20151
settheory&algebra
normal
lattice
+41
votes
5
answers
10
GATE2015116
For a set $A$, the power set of $A$ is denoted by $2^{A}$. If $A = \left\{5,\left\{6\right\}, \left\{7\right\}\right\}$, which of the following options are TRUE? $\phi \in 2^{A}$ $\phi \subseteq 2^{A}$ $\left\{5,\left\{6\right\}\right\} \in 2^{A}$ $\left\{5,\left\{6\right\}\right\} \subseteq 2^{A}$ I and III only II and III only I, II and III only I, II and IV only
asked
Feb 13, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
30.8k
points)

5.6k
views
gate20151
settheory&algebra
sets
normal
+39
votes
4
answers
11
GATE201432
Let $X$ and $Y$ be finite sets and $f:X \to Y$ be a function. Which one of the following statements is TRUE? For any subsets $A$ and $B$ of $X, fA \cup B = f(A) + f(B)$ For any subsets $A$ and $B$ of $X, f(A \cap B) = f(A) \cap f(B)$ For any subsets $A$ and $B$ ... $S$ and $T$ of $Y, f^{1}(S \cap T) = f^{1}(S) \cap f^{1}(T)$
asked
Sep 28, 2014
in
Set Theory & Algebra
by
jothee
Veteran
(
106k
points)

4.5k
views
gate20143
settheory&algebra
functions
normal
+39
votes
5
answers
12
GATE20002.6
Let $P(S)$ denotes the power set of set $S.$ Which of the following is always true? $P(P(S)) = P(S)$ $P(S) ∩ P(P(S)) = \{ Ø \}$ $P(S) ∩ S = P(S)$ $S ∉ P(S)$
asked
Sep 14, 2014
in
Set Theory & Algebra
by
Kathleen
Veteran
(
52.2k
points)

4.3k
views
gate2000
settheory&algebra
easy
sets
+37
votes
4
answers
13
GATE200337
Let \(f : A \to B\) be an injective (onetoone) 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 subsets $D$ ... is always true? \(g(h(D)) \subseteq D\) \(g(h(D)) \supseteq D\) \(g(h(D)) \cap D = \phi\) \(g(h(D)) \cap (B  D) \ne \phi\)
asked
Sep 16, 2014
in
Set Theory & Algebra
by
Kathleen
Veteran
(
52.2k
points)

2.9k
views
gate2003
settheory&algebra
functions
normal
+34
votes
10
answers
14
GATE2015240
The number of onto functions (surjective functions) from set $X = \{1, 2, 3, 4\}$ to set $Y=\{a,b,c\}$ is ______.
asked
Feb 13, 2015
in
Set Theory & Algebra
by
jothee
Veteran
(
106k
points)

8.3k
views
gate20152
settheory&algebra
functions
normal
numericalanswers
+34
votes
5
answers
15
GATE200624
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 integers obtained by applying ... $n! \frac{A ∩ B}{A ∪ B}$ $\dfrac{A ∩ B^2}{^n \mathrm{C}_{A ∪ B}}$
asked
Sep 18, 2014
in
Set Theory & Algebra
by
Rucha Shelke
Active
(
3.3k
points)

3.3k
views
gate2006
settheory&algebra
normal
sets
+34
votes
3
answers
16
GATE200331
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 $x \leq y$ ... False 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
asked
Sep 16, 2014
in
Set Theory & Algebra
by
Kathleen
Veteran
(
52.2k
points)

3.4k
views
gate2003
settheory&algebra
partialorder
normal
propositionallogic
+33
votes
6
answers
17
GATE2016226
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? Both $P$ and $Q$ are true. $P$ is true and $Q$ is false. $P$ is false and $Q$ is true. Both $P$ and $Q$ are false.
asked
Feb 12, 2016
in
Set Theory & Algebra
by
Akash Kanase
Boss
(
41.9k
points)

4.9k
views
gate20162
settheory&algebra
relations
normal
+33
votes
5
answers
18
GATE2015341
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 $ps=qr$. 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
asked
Feb 15, 2015
in
Set Theory & Algebra
by
jothee
Veteran
(
106k
points)

4.1k
views
gate20153
settheory&algebra
relations
normal
+33
votes
4
answers
19
GATE20012.2
Consider the following statements: $S1:$ There exists infinite sets $A$, $B$, $C$ such that $A \cap (B \cup C)$ is finite. $S2:$ There exists two irrational numbers $x$ and y such that $(x+y)$ is rational. Which of the following is true about $S1$ and $S2$? Only $S1$ is correct Only $S2$ is correct Both $S1$ and $S2$ are correct None of $S1$ and $S2$ is correct
asked
Sep 14, 2014
in
Set Theory & Algebra
by
Kathleen
Veteran
(
52.2k
points)

2.5k
views
gate2001
settheory&algebra
normal
sets
+32
votes
3
answers
20
GATE2015323
Suppose $U$ is the power set of the set $S = \{1, 2, 3, 4, 5, 6\}$. For any $T \in U$, let $T$ denote the number of elements in $T$ and $T'$ denote the complement of $T$. For any $T, R \in U \text{ let } T \backslash R$ be the set of all elements in $T$ which ... $X \backslash Y = \phi)$ $\forall X \in U, \forall Y \in U, (X \backslash Y = Y' \backslash X')$
asked
Feb 14, 2015
in
Set Theory & Algebra
by
jothee
Veteran
(
106k
points)

3.9k
views
gate20153
settheory&algebra
sets
normal
+32
votes
4
answers
21
GATE19976.3
The number of equivalence relations of the set $\{1,2,3,4\}$ is $15$ $16$ $24$ $4$
asked
Sep 29, 2014
in
Set Theory & Algebra
by
Kathleen
Veteran
(
52.2k
points)

8.5k
views
gate1997
settheory&algebra
relations
normal
+31
votes
5
answers
22
GATE20006
Let $S$ be a set of $n$ elements $\left\{1, 2,....., n\right\}$ and $G$ a graph with 2$^{n}$ vertices, each vertex corresponding to a distinct subset of $S$. Two vertices are adjacent iff the symmetric difference of the corresponding sets has exactly $2$ ... $G$ has the same degree. What is the degree of a vertex in $G$? How many connected components does $G$ have?
asked
Sep 14, 2014
in
Set Theory & Algebra
by
Kathleen
Veteran
(
52.2k
points)

1.9k
views
gate2000
settheory&algebra
normal
descriptive
sets
+30
votes
4
answers
23
GATE201237
How many onto (or surjective) functions are there from an $n$element $(n ≥ 2)$ set to a $2$element set? $ 2^{n}$ $2^{n} – 1$ $2^{n} – 2$ $2(2^{n} – 2)$
asked
Sep 26, 2014
in
Set Theory & Algebra
by
gatecse
Boss
(
17.5k
points)

2.7k
views
gate2012
settheory&algebra
functions
normal
+30
votes
3
answers
24
GATE20012.3
Let $f: A \rightarrow B$ a function, and let E and F be subsets of $A$. Consider the following statements about images. $S1: f(E \cup F) = f(E) \cup f(F)$ $S2: f(E \cap F)=f(E) \cap f(F)$ Which of the following is true about S1 and S2? Only S1 is correct Only S2 is correct Both S1 and S2 are correct None of S1 and S2 is correct
asked
Sep 14, 2014
in
Set Theory & Algebra
by
Kathleen
Veteran
(
52.2k
points)

3.4k
views
gate2001
settheory&algebra
functions
normal
+29
votes
1
answer
25
GATE19891v
The number of possible commutative binary operations that can be defined on a set of $n$ elements (for a given n) is ___________.
asked
Nov 27, 2016
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
30.8k
points)

1.9k
views
gate1989
descriptive
settheory&algebra
binaryoperation
+29
votes
2
answers
26
TIFR2012A8
How many pairs of sets $(A, B)$ are there that satisfy the condition $A, B \subseteq \left\{1, 2,...,5\right\}, A \cap B = \{\}?$ $125$ $127$ $243$ $257$
asked
Oct 26, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
30.8k
points)

781
views
tifr2012
settheory&algebra
sets
+29
votes
4
answers
27
GATE2015254
Let $X$ and $Y$ denote the sets containing 2 and 20 distinct objects respectively and $F$ denote the set of all possible functions defined from $X$ to $Y$. Let $f$ be randomly chosen from $F$. The probability of $f$ being onetoone is ______.
asked
Feb 13, 2015
in
Set Theory & Algebra
by
jothee
Veteran
(
106k
points)

2.8k
views
gate20152
settheory&algebra
functions
normal
numericalanswers
+29
votes
3
answers
28
GATE2006IT2
For the set $N$ of natural numbers and a binary operation $f : N \times N \to N,$ an element $z \in N$ is called an identity for $f,$ if $f (a, z) = a = f(z, a),$ for all $a \in N.$ Which of the following binary operations have an identity? $f (x, y) = x + y  3$ $f (x, y) = \max(x, y)$ $f (x, y) = x^y$ I and II only II and III only I and III only None of these
asked
Oct 31, 2014
in
Set Theory & Algebra
by
Ishrat Jahan
Boss
(
16.3k
points)

2.5k
views
gate2006it
settheory&algebra
easy
binaryoperation
+28
votes
5
answers
29
GATE201827
Let $N$ be the set of natural numbers. Consider the following sets, $P:$ Set of Rational numbers (positive and negative) $Q:$ Set of functions from $\{0,1\}$ to $N$ $R:$ Set of functions from $N$ to $\{0, 1\}$ $S:$ Set of finite subsets of $N$ Which of the above sets are countable? $Q$ and $S$ only $P$ and $S$ only $P$ and $R$ only $P, Q$ and $S$ only
asked
Feb 14, 2018
in
Set Theory & Algebra
by
gatecse
Boss
(
17.5k
points)

5.7k
views
gate2018
settheory&algebra
countableuncountableset
normal
+28
votes
3
answers
30
GATE19942.2
On the set $N$ of nonnegative integers, the binary operation ______ is associative and noncommutative.
asked
Oct 4, 2014
in
Set Theory & Algebra
by
Kathleen
Veteran
(
52.2k
points)

1.3k
views
gate1994
settheory&algebra
normal
binaryoperation
descriptive
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