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Recent questions and answers in Set Theory & Algebra
+58
votes
7
answers
1
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 ___________.
answered
1 day
ago
in
Set Theory & Algebra
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JashanArora
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6.9k
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6.8k
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gate20161
settheory&algebra
functions
normal
numericalanswers
+55
votes
8
answers
2
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.
answered
4 days
ago
in
Set Theory & Algebra
by
King Suleiman
Active
(
1.2k
points)

5.6k
views
gate20162
settheory&algebra
difficult
sets
+41
votes
6
answers
3
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 $x, y$ ... $p_r = 0$ $p_r = 1$ $0 < p_r ≤ \frac{1}{5}$ $\frac{1}{5} < p_r < 1$
answered
4 days
ago
in
Set Theory & Algebra
by
JashanArora
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6.9k
points)

5.9k
views
gate20151
settheory&algebra
normal
lattice
+41
votes
5
answers
4
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
answered
5 days
ago
in
Set Theory & Algebra
by
JashanArora
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6.9k
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5.5k
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gate20151
settheory&algebra
sets
normal
+50
votes
6
answers
5
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: $3m$ $3n$ $2m+1$ $2n+1$
answered
Jan 18
in
Set Theory & Algebra
by
Madhab
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(
5.8k
points)

3k
views
gate2006
settheory&algebra
normal
functions
+33
votes
6
answers
6
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.
answered
Jan 18
in
Set Theory & Algebra
by
arjuno
(
299
points)

4.9k
views
gate20162
settheory&algebra
relations
normal
+1
vote
1
answer
7
ISRO202076
If $A=\{x,y,z\}$ and $B=\{u,v,w,x\}, $ and the universe is $\{s,t,u,v,w,x,y,z\}$ Then $(A \cup B’) \cap (A \cap B)$ is equal to $\{u,v,w,x\}$ $\{ \ \}$ $\{u,v,w,x,y,z\}$ $\{u,v,w\}$
answered
Jan 13
in
Set Theory & Algebra
by
habedo007
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2.8k
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141
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isro2020
discretemathematics
settheory&algebra
sets
easy
+2
votes
1
answer
8
GO2019FLT111
Which one of the following best expresses the generating function sequence $\{a_n\}$, for the given closed form representation? $F(x) = \frac{1}{1xx^2}$ $a_n=a_{n1}+3, n>0, a_0=1$ $a_n=a_{n1}+a_{n2}, n>1, a_0=1, a_1=1$ $a_n=2n+3, n>1$ $a_n=2a_{n1}+3, n>1, a_0=1$
answered
Jan 12
in
Set Theory & Algebra
by
blackcloud
Junior
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553
points)

326
views
go2019flt1
generatingfunctions
settheory&algebra
+22
votes
6
answers
9
GATE2005IT34
Let $n =$ $p^{2}q$, where $p$ and $q$ are distinct prime numbers. How many numbers m satisfy $1 ≤ m ≤ n$ and $gcd$ $(m, n) = 1?$ Note that $gcd$ $(m, n)$ is the greatest common divisor of $m$ and $n$. $p(q  1)$ $pq$ $\left ( p^{2}1 \right ) (q  1)$ $p(p  1) (q  1)$
answered
Jan 9
in
Set Theory & Algebra
by
arjuno
(
299
points)

2.1k
views
gate2005it
settheory&algebra
normal
numbertheory
+29
votes
4
answers
10
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 ______.
answered
Jan 4
in
Set Theory & Algebra
by
shivam001
Junior
(
987
points)

2.8k
views
gate20152
settheory&algebra
functions
normal
numericalanswers
+24
votes
3
answers
11
GATE19962.1
Let $R$ denote the set of real numbers. Let $f:R\times R \rightarrow R \times R$ be a bijective function defined by $f(x,y) = (x+y, xy)$. The inverse function of $f$ is given by $f^{1} (x,y) = \left( \frac {1}{x+y}, \frac{1}{xy}\right)$ ... $f^{1}(x,y)=\left [ 2\left(xy\right),2\left(x+y\right) \right ]$
answered
Jan 4
in
Set Theory & Algebra
by
shivam001
Junior
(
987
points)

2.7k
views
gate1996
settheory&algebra
functions
normal
+2
votes
1
answer
12
ISI 2004 MIII
Let $X$ be a nonempty set and let $\mathcal{P}(X)$ denote the collection of all subsets of $X.$ Define $\textit{f}:\textit{X$\times$ $\mathcal{P}$(X)}\rightarrow \mathbb{R}$ by $f(x,A) = \begin{cases} 1 \text{ if } x \in A & \\ 0 \text{ if } x \notin A & \end{cases}$ ... $f(x,A)+f(x,B)  f(x,A) \cdot f(x,B)$ $f(x,A)+ \mid f(x,A)  f(x,B) \mid$
answered
Dec 31, 2019
in
Set Theory & Algebra
by
ankitgupta.1729
Boss
(
17.1k
points)

156
views
isi2004
functions
0
votes
1
answer
13
$\textbf{NTA NET DEC 2019 (group)}$
Consider the following statements: $\mathbf{S_1:}\;\;$If a group $\mathbf{(G,*)}$ is of order $\mathbf n$ and $\mathrm {a \in G}$ is such that $\mathrm {a^n=e}$ for some integer $\mathrm {m \le n}$ then $\mathbf m$ must divide $\mathbf n$ ... $(3)\;\;\;\text{Niether}\; \mathrm{S_1}\;\text{nor}\;\mathrm{S_2}$
answered
Dec 30, 2019
in
Set Theory & Algebra
by
`JEET
Boss
(
19.2k
points)

100
views
ugcnetdec2019ii
grouptheory
+2
votes
1
answer
14
GATE19879f
Give the composition tables (Cayley Tables) of the two non isomorphic groups of order 4 with elements $e, a, b, c$ where $c$ is the identity element. Use the order $e, a, b, c$ for the rows and columns.
answered
Dec 30, 2019
in
Set Theory & Algebra
by
ankitgupta.1729
Boss
(
17.1k
points)

217
views
gate1987
groupisomorphism
settheory&algebra
nongate
+59
votes
10
answers
15
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
answered
Dec 29, 2019
in
Set Theory & Algebra
by
pranshu27
(
21
points)

8.4k
views
gate20151
settheory&algebra
functions
difficult
0
votes
2
answers
16
ISI2016MMA13
Which one of the following statements is correct regarding the elements and subsets of the set $\{1, 2, \{1, 2, 3\}\}$? $\{1, 2\} \in \{1, 2, \{1, 2, 3\} \}$ $\{1, 2\} \subseteq \{1, 2, \{1, 2, 3\} \}$ $\{1, 2, 3\} \subseteq \{1, 2, \{1, 2, 3\} \}$ $3 \in \{1, 2, \{1, 2, 3\} \}$
answered
Dec 26, 2019
in
Set Theory & Algebra
by
smsubham
Boss
(
11.8k
points)

25
views
isi2016mmamma
sets
subsets
0
votes
2
answers
17
Hasse Doubt
what is the least upper bound of {a, b, c}?
answered
Dec 26, 2019
in
Set Theory & Algebra
by
spike500
(
41
points)

135
views
hassediagram
settheory&algebra
lattice
partialorder
+1
vote
3
answers
18
PGEE 2018
let 5,8,11,14,17,20.. be series then 320 will be which term of this series A) 104 B) 106 C) 962 D) 87
answered
Dec 26, 2019
in
Set Theory & Algebra
by
Yash4444
Junior
(
841
points)

336
views
iiithpgee
+10
votes
4
answers
19
GATE19871xxii
The equation $7x^{7}+14x^{6}+12x^{5}+3x^{4}+12x^{3}+10x^{2}+5x+7=0$ has All complex roots At least one real root Four pairs of imaginary roots None of the above
answered
Dec 22, 2019
in
Set Theory & Algebra
by
pritishc
Active
(
2.2k
points)

527
views
gate1987
polynomials
+18
votes
3
answers
20
GATE200546
Consider the set $H$ of all $3 * 3$ matrices of the type $\left( \begin{array}{ccc} a & f & e \\ 0 & b & d \\ 0 & 0 & c \end{array} \right)$ where $a,b,c,d,e$ and $f$ are real numbers and $abc ≠ 0.$ Under the matrix multiplication operation, the set $H$ is: a group a monoid but not a group a semi group but not a monoid neither a group nor a semi group
answered
Dec 19, 2019
in
Set Theory & Algebra
by
JashanArora
Loyal
(
6.9k
points)

2.2k
views
gate2005
settheory&algebra
grouptheory
normal
+23
votes
3
answers
21
GATE199214a
If $G$ is a group of even order, then show that there exists an element $a≠e$, $e$, the identity in $G$, such that $a^2 = e$.
answered
Dec 19, 2019
in
Set Theory & Algebra
by
JashanArora
Loyal
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6.9k
points)

1.2k
views
gate1992
settheory&algebra
grouptheory
normal
descriptive
proof
+22
votes
2
answers
22
GATE20037
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 noncommutative group does not have a right identity element forms a group if the empty string is removed from $\Sigma^*$
answered
Dec 19, 2019
in
Set Theory & Algebra
by
JashanArora
Loyal
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6.9k
points)

2.4k
views
gate2003
settheory&algebra
grouptheory
normal
+15
votes
6
answers
23
GATE19973.1
Let $\left(Z, *\right)$ be an algebraic structure where $Z$ is the set of integers and the operation $*$ is defined by $n*m = \max(n,m)$. Which of the following statements is true for $\left(Z, *\right)$? $\left(Z, *\right)$ is a monoid $\left(Z, *\right)$ is an Abelian group $\left(Z, *\right)$ is a group None of the above
answered
Dec 19, 2019
in
Set Theory & Algebra
by
JashanArora
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6.9k
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2k
views
gate1997
settheory&algebra
grouptheory
normal
+19
votes
3
answers
24
GATE19961.4
Which of the following statements is FALSE? The set of rational numbers is an abelian group under addition The set of integers in an abelian group under addition The set of rational numbers form an abelian group under multiplication The set of real numbers excluding zero is an abelian group under multiplication
answered
Dec 19, 2019
in
Set Theory & Algebra
by
JashanArora
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6.9k
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2.1k
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gate1996
settheory&algebra
grouptheory
normal
+18
votes
3
answers
25
GATE19952.17
Let $A$ be the set of all nonsingular 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.
answered
Dec 19, 2019
in
Set Theory & Algebra
by
JashanArora
Loyal
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6.9k
points)

1.7k
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gate1995
settheory&algebra
grouptheory
+25
votes
2
answers
26
GATE19941.10
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
answered
Dec 19, 2019
in
Set Theory & Algebra
by
JashanArora
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6.9k
points)

2.8k
views
gate1994
settheory&algebra
grouptheory
normal
0
votes
3
answers
27
Distributive lattice
Q)which of the following is not a distributive lattice? a) [P(A);$\preceq$ ] where A = { a,b,c,d } b) [ {1,2,3,5,30} ; / ]
answered
Dec 19, 2019
in
Set Theory & Algebra
by
JashanArora
Loyal
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6.9k
points)

407
views
discretemathematics
settheory&algebra
lattice
+15
votes
3
answers
28
GATE20059
The following is the Hasse diagram of the poset $\left[\{a,b,c,d,e\},≺\right]$ The poset is : not a lattice a lattice but not a distributive lattice a distributive lattice but not a Boolean algebra a Boolean algebra
answered
Dec 19, 2019
in
Set Theory & Algebra
by
JashanArora
Loyal
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6.9k
points)

2k
views
gate2005
settheory&algebra
lattice
normal
+15
votes
3
answers
29
GATE19973.3
In the lattice defined by the Hasse diagram given in following figure, how many complements does the element ‘$e$’ have? $2$ $3$ $0$ $1$
answered
Dec 19, 2019
in
Set Theory & Algebra
by
JashanArora
Loyal
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6.9k
points)

1.9k
views
gate1997
settheory&algebra
lattice
normal
+11
votes
2
answers
30
GATE19881vii
The complement(s) of the element 'a' in the lattice shown in below figure is (are) ____
answered
Dec 19, 2019
in
Set Theory & Algebra
by
JashanArora
Loyal
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6.9k
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1k
views
gate1988
descriptive
lattice
settheory&algebra
+9
votes
4
answers
31
GATE19942.9
The Hasse diagrams of all the lattices with up to four elements are _____ (write all the relevant Hasse diagrams)
answered
Dec 18, 2019
in
Set Theory & Algebra
by
JashanArora
Loyal
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6.9k
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1k
views
gate1994
settheory&algebra
lattice
normal
descriptive
+13
votes
4
answers
32
TIFR2014B16
Consider the ordering relation $x\mid y \subseteq N \times N$ over natural numbers $N$ such that $x \mid y$ if there exists $z \in N$ such that $x ∙ z = y$. A set is called lattice if every finite subset has a least upper bound and greatest lower bound. It ... $\mid$ is a total order. $(N, \mid)$ is a complete lattice. $(N, \mid)$ is a lattice but not a complete lattice.
answered
Dec 18, 2019
in
Set Theory & Algebra
by
JashanArora
Loyal
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6.9k
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991
views
tifr2014
settheory&algebra
partialorder
+13
votes
3
answers
33
TIFR2014B15
Consider the set $N^{*}$ of finite sequences of natural numbers with $x \leq_{p}y$ denoting that sequence $x$ is a prefix of sequence $y$. Then, which of the following is true? $N^{*}$ is uncountable. $\leq_{p}$ is a total order. Every nonempty subset of ... Every nonempty subset of $N^{*}$ has a greatest lower bound. Every nonempty finite subset of $N^{*}$ has a least upper bound.
answered
Dec 18, 2019
in
Set Theory & Algebra
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JashanArora
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6.9k
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764
views
tifr2014
settheory&algebra
partialorder
+9
votes
3
answers
34
TIFR2012B5
Let $R$ be a binary relation over a set $S$. The binary relation $R$ ... $\sqsubseteq $ is an equivalence relation and a well order. $\sqsubseteq $ is neither a partial order nor an equivalence relation.
answered
Dec 18, 2019
in
Set Theory & Algebra
by
JashanArora
Loyal
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6.9k
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530
views
tifr2012
settheory&algebra
partialorder
+21
votes
6
answers
35
GATE2007IT23
A partial order $P$ is defined on the set of natural numbers as follows. Here $\frac{x}{y}$ denotes integer division. $(0, 0) \in P.$ $(a, b) \in P$ if and only if $(a \% 10) \leq (b \% 10$) and $(\frac{a}{10},\frac{b}{10})\in P.$ Consider the ... $P$? (i) and (iii) (ii) and (iv) (i) and (iv) (iii) and (iv)
answered
Dec 18, 2019
in
Set Theory & Algebra
by
JashanArora
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6.9k
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2.6k
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gate2007it
settheory&algebra
partialorder
normal
+34
votes
3
answers
36
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
answered
Dec 18, 2019
in
Set Theory & Algebra
by
JashanArora
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6.9k
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3.4k
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gate2003
settheory&algebra
partialorder
normal
propositionallogic
+12
votes
3
answers
37
GATE199101,xiv
If the longest chain in a partial order is of length $n$, then the partial order can be written as a _____ of $n$ antichains.
answered
Dec 18, 2019
in
Set Theory & Algebra
by
JashanArora
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6.9k
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gate1991
settheory&algebra
partialorder
normal
descriptive
+28
votes
2
answers
38
GATE19976.1
A partial order $≤$ is defined on the set $S=\left \{ x, a_1, a_2, \ldots, a_n, y \right \}$ as $x$ $\leq _{i}$ $a_{i}$ for all $i$ and $a_{i}\leq y$ for all $i$, where $n ≥ 1$. The number of total orders on the set S which contain the partial order $≤$ is $n!$ $n+2$ $n$ $1$
answered
Dec 18, 2019
in
Set Theory & Algebra
by
JashanArora
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6.9k
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2.6k
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gate1997
settheory&algebra
partialorder
normal
+20
votes
4
answers
39
GATE199811
Suppose $A = \{a, b, c, d\}$ and $\Pi_1$ is the following partition of A $\Pi_1 = \left\{\left\{a, b, c\right\}\left\{d\right\}\right\}$ List the ordered pairs of the equivalence relations induced by $\Pi_1$. Draw the graph of the above equivalence relation ... $\left\langle\left\{\Pi_1, \Pi_2, \Pi_3, \Pi_4\right\}, \text{ refines } \right\rangle$.
answered
Dec 18, 2019
in
Set Theory & Algebra
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JashanArora
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6.9k
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2.2k
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gate1998
settheory&algebra
normal
partialorder
descriptive
+10
votes
2
answers
40
ISRODEC20172
Consider the set of integers $I.$ Let $D$ denote "divides with an integer quotient" (e.g. $4D8$ but not $4D7$). Then $D$ is Reflexive, Not Symmetric, Transitive Not Reflexive, Not Antisymmetric, Transitive Reflexive, Antisymmetric, Transitive Not Reflexive, Not Antisymmetric, Not Transitive
answered
Dec 11, 2019
in
Set Theory & Algebra
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JashanArora
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2.6k
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isrodec2017
settheory&algebra
relations
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