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Recent questions and answers in Discrete Mathematics
37
votes
5
answers
1
GATE2001-2.15
How many undirected graphs (not necessarily connected) can be constructed out of a given set $V=\{v_1, v_2, \dots v_n\}$ of $n$ vertices? $\frac{n(n-1)} {2}$ $2^n$ $n!$ $2^\frac{n(n-1)} {2} $
How many undirected graphs (not necessarily connected) can be constructed out of a given set $V=\{v_1, v_2, \dots v_n\}$ of $n$ vertices? $\frac{n(n-1)} {2}$ $2^n$ $n!$ $2^\frac{n(n-1)} {2} $
answered
16 hours
ago
in
Graph Theory
varunrajarathnam
6.2k
views
gate2001
graph-theory
normal
counting
17
votes
5
answers
2
GATE2009-26
Consider the following well-formed formulae: $\neg \forall x(P(x))$ $\neg \exists x(P(x))$ $\neg \exists x(\neg P(x))$ $\exists x(\neg P(x))$ Which of the above are equivalent? I and III I and IV II and III II and IV
Consider the following well-formed formulae: $\neg \forall x(P(x))$ $\neg \exists x(P(x))$ $\neg \exists x(\neg P(x))$ $\exists x(\neg P(x))$ Which of the above are equivalent? I and III I and IV II and III II and IV
answered
1 day
ago
in
Mathematical Logic
Adarsh Pandey
1.8k
views
gate2009
mathematical-logic
normal
first-order-logic
1
vote
4
answers
3
NIELIT 2017 DEC Scientist B - Section B: 52
Let $G$ be a simple undirected graph on $n=3x$ vertices $(x \geq 1)$ with chromatic number $3$, then maximum number of edges in $G$ is $n(n-1)/2$ $n^{n-2}$ $nx$ $n$
Let $G$ be a simple undirected graph on $n=3x$ vertices $(x \geq 1)$ with chromatic number $3$, then maximum number of edges in $G$ is $n(n-1)/2$ $n^{n-2}$ $nx$ $n$
answered
1 day
ago
in
Graph Theory
debasish paramanik
468
views
nielit2017dec-scientistb
discrete-mathematics
graph-theory
graph-coloring
15
votes
4
answers
4
GATE2019-10
Let $G$ be an arbitrary group. Consider the following relations on $G$: $R_1: \forall a , b \in G, \: a R_1 b \text{ if and only if } \exists g \in G \text{ such that } a = g^{-1}bg$ $R_2: \forall a , b \in G, \: a R_2 b \text{ if and only if } a= b^{-1}$ Which of the above is/are equivalence relation/relations? $R_1$ and $R_2$ $R_1$ only $R_2$ only Neither $R_1$ nor $R_2$
Let $G$ be an arbitrary group. Consider the following relations on $G$: $R_1: \forall a , b \in G, \: a R_1 b \text{ if and only if } \exists g \in G \text{ such that } a = g^{-1}bg$ $R_2: \forall a , b \in G, \: a R_2 b \text{ if and only if } a= b^{-1}$ Which of the above is/are equivalence relation/relations? $R_1$ and $R_2$ $R_1$ only $R_2$ only Neither $R_1$ nor $R_2$
answered
3 days
ago
in
Set Theory & Algebra
Amcodes
5.5k
views
gate2019
engineering-mathematics
discrete-mathematics
set-theory&algebra
group-theory
14
votes
4
answers
5
TIFR2015-A-5
What is logically equivalent to "If Kareena and Parineeti go to the shopping mall then it is raining": If Kareena and Parineeti do not go to the shopping mall then it is not raining. If Kareena and Parineeti do not go to the shopping mall then it is ... to the shopping mall. If it is not raining then Kareena and Parineeti do not go to the shopping mall. None of the above.
What is logically equivalent to "If Kareena and Parineeti go to the shopping mall then it is raining": If Kareena and Parineeti do not go to the shopping mall then it is not raining. If Kareena and Parineeti do not go to the shopping mall then it is raining. If it ... go to the shopping mall. If it is not raining then Kareena and Parineeti do not go to the shopping mall. None of the above.
answered
5 days
ago
in
Mathematical Logic
iamalokpandey
775
views
tifr2015
mathematical-logic
propositional-logic
31
votes
2
answers
6
GATE2005-IT-46
A line $L$ in a circuit is said to have a $stuck-at-0$ fault if the line permanently has a logic value $0$. Similarly a line $L$ in a circuit is said to have a $stuck-at-1$ fault if the line permanently has a logic value $1$ ... number of distinct multiple $stuck-at$ faults possible in a circuit with $N$ lines is $3^N$ $3^N - 1$ $2^N - 1$ $2$
A line $L$ in a circuit is said to have a $stuck-at-0$ fault if the line permanently has a logic value $0$. Similarly a line $L$ in a circuit is said to have a $stuck-at-1$ fault if the line permanently has a logic value $1$. A circuit is said to have a multiple $stuck-at$ ... total number of distinct multiple $stuck-at$ faults possible in a circuit with $N$ lines is $3^N$ $3^N - 1$ $2^N - 1$ $2$
answered
Oct 17
in
Combinatory
varunrajarathnam
2.7k
views
gate2005-it
combinatory
normal
30
votes
3
answers
7
GATE2003-5
$n$ 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 number of different gatherings possible at the party is \(^{2n}\mathrm{C}_n\times 2^n\) \(3^n\) \(\frac{(2n)!}{2^n}\) \(^{2n}\mathrm{C}_n\)
answered
Oct 16
in
Combinatory
varunrajarathnam
3.7k
views
gate2003
combinatory
normal
34
votes
4
answers
8
GATE2003-4
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 that each is sorted in ascending order, $B$ has $5$ and $C$ has $3$ elements, and the result of merging $B$ and $C$ gives $A$ $2$ $30$ $56$ $256$
answered
Oct 16
in
Combinatory
varunrajarathnam
5.2k
views
gate2003
combinatory
normal
25
votes
5
answers
9
GATE1999-2.2
Two girls have picked $10$ roses, $15$ sunflowers and $15$ daffodils. What is the number of ways they can divide the flowers among themselves? $1638$ $2100$ $2640$ None of the above
Two girls have picked $10$ roses, $15$ sunflowers and $15$ daffodils. What is the number of ways they can divide the flowers among themselves? $1638$ $2100$ $2640$ None of the above
answered
Oct 14
in
Combinatory
varunrajarathnam
5.6k
views
gate1999
combinatory
normal
23
votes
5
answers
10
GATE1999-1.3
The number of binary strings of $n$ zeros and $k$ ones in which no two ones are adjacent is $^{n-1}C_k$ $^nC_k$ $^nC_{k+1}$ None of the above
The number of binary strings of $n$ zeros and $k$ ones in which no two ones are adjacent is $^{n-1}C_k$ $^nC_k$ $^nC_{k+1}$ None of the above
answered
Oct 14
in
Combinatory
varunrajarathnam
3.5k
views
gate1999
combinatory
normal
17
votes
3
answers
11
GATE1998-1.23
How many sub strings of different lengths (non-zero) can be formed from a character string of length $n$? $n$ $n^2$ $2^n$ $\frac{n(n+1)}{2}$
How many sub strings of different lengths (non-zero) can be formed from a character string of length $n$? $n$ $n^2$ $2^n$ $\frac{n(n+1)}{2}$
answered
Oct 14
in
Combinatory
varunrajarathnam
6.1k
views
gate1998
combinatory
normal
19
votes
4
answers
12
GATE1994-1.15
The number of substrings (of all lengths inclusive) that can be formed from a character string of length $n$ is $n$ $n^2$ $\frac{n(n-1)}{2}$ $\frac{n(n+1)}{2}$
The number of substrings (of all lengths inclusive) that can be formed from a character string of length $n$ is $n$ $n^2$ $\frac{n(n-1)}{2}$ $\frac{n(n+1)}{2}$
answered
Oct 14
in
Combinatory
varunrajarathnam
3.3k
views
gate1994
combinatory
normal
4
votes
2
answers
13
Graph Theory conceptual
A simple graph is one in which there are no self loops and each pair of distinct vertices is connected by at most one edge. Let G be a simple graph on 8 vertices such that there is a vertex of degree 1, a vertex of degree 2, a vertex of degree 3, a vertex of degree ... a vertex of degree 7. Which of the following can be the degree of the last vertex? (A) 3 (B) 0 (C) 5 (D) 4
A simple graph is one in which there are no self loops and each pair of distinct vertices is connected by at most one edge. Let G be a simple graph on 8 vertices such that there is a vertex of degree 1, a vertex of degree 2, a vertex of degree 3, a vertex of degree 4, a vertex of ... 6 and a vertex of degree 7. Which of the following can be the degree of the last vertex? (A) 3 (B) 0 (C) 5 (D) 4
answered
Oct 14
in
Graph Theory
ayush.5
201
views
graph-theory
discrete-mathematics
0
votes
2
answers
14
Gateforum Test Series: Graph Theory - Graph Matching
answered
Oct 14
in
Graph Theory
wander
183
views
gateforum-test-series
discrete-mathematics
graph-theory
graph-matching
0
votes
2
answers
15
ME test series question on graph theory
answered
Oct 13
in
Graph Theory
arun yadav
144
views
graph-theory
17
votes
4
answers
16
TIFR2017-A-5
How many distinct ways are there to split $50$ identical coins among three people so that each person gets at least $5$ coins? $3^{35}$ $3^{50}-2^{50}$ $\binom{35}{2}$ $\binom{50}{15} \cdot 3^{35}$ $\binom{37}{2}$
How many distinct ways are there to split $50$ identical coins among three people so that each person gets at least $5$ coins? $3^{35}$ $3^{50}-2^{50}$ $\binom{35}{2}$ $\binom{50}{15} \cdot 3^{35}$ $\binom{37}{2}$
answered
Oct 13
in
Combinatory
varunrajarathnam
1.7k
views
tifr2017
combinatory
discrete-mathematics
normal
balls-in-bins
0
votes
2
answers
17
Zeal Test Series 2019: Graph Theory - Graph Connectivity
answered
Oct 13
in
Graph Theory
arun yadav
232
views
zeal
graph-theory
graph-connectivity
zeal2019
1
vote
2
answers
18
Made easy Test Series:Graph Theory+Automata
Consider a graph $G$ with $2^{n}$ vertices where the level of each vertex is a $n$ bit binary string represented as $a_{0},a_{1},a_{2},.............,a_{n-1}$, where each $a_{i}$ is $0$ or $1$ ... and $y$ denote the degree of a vertex $G$ and number of connected component of $G$ for $n=8.$ The value of $x+10y$ is_____________
Consider a graph $G$ with $2^{n}$ vertices where the level of each vertex is a $n$ bit binary string represented as $a_{0},a_{1},a_{2},.............,a_{n-1}$, where each $a_{i}$ is $0$ or $1$ ... $x$ and $y$ denote the degree of a vertex $G$ and number of connected component of $G$ for $n=8.$ The value of $x+10y$ is_____________
answered
Oct 13
in
Graph Theory
arun yadav
287
views
made-easy-test-series
graph-theory
theory-of-computation
2
votes
2
answers
19
testbook
Which of the above lattice is distributve? a) both iii and iv) b) only iv)
Which of the above lattice is distributve? a) both iii and iv) b) only iv)
answered
Oct 12
in
Set Theory & Algebra
arun yadav
108
views
0
votes
2
answers
20
Lattice
Answer given is option C , But vertex 10 do not have compliment then how it can be a Boolean algebra ? Also please explain , as no element has compliment greater than 1 , it may or may not be distributive then is there any feasible way to differentiate between option a and d ? Thank you ! P.S : Without using distributive law check, i think its not feasible for more number of vertices.
Answer given is option C , But vertex 10 do not have compliment then how it can be a Boolean algebra ? Also please explain , as no element has compliment greater than 1 , it may or may not be distributive then is there any feasible way to differentiate between option a and d ? Thank you ! P.S : Without using distributive law check, i think its not feasible for more number of vertices.
answered
Oct 12
in
Set Theory & Algebra
arun yadav
222
views
28
votes
7
answers
21
GATE2009-22
For the composition table of a cyclic group shown below: ... $a,b$ are generators $b,c$ are generators $c,d$ are generators $d,a$ are generators
For the composition table of a cyclic group shown below: ... $a,b$ are generators $b,c$ are generators $c,d$ are generators $d,a$ are generators
answered
Oct 12
in
Set Theory & Algebra
Lasani Hussain
3.1k
views
gate2009
set-theory&algebra
normal
group-theory
0
votes
2
answers
22
MadeEasy Test Series: Set Theory & Algebra - Relations
answered
Oct 12
in
Mathematical Logic
ayush.5
131
views
made-easy-test-series
set-theory&algebra
relations
1
vote
2
answers
23
relations and functions
answered
Oct 12
in
Mathematical Logic
arun yadav
97
views
2
votes
1
answer
24
Group theory
Let the number of non-isomorphic groups of order 10 be X and number of non-isomorphic groups of order 24 be Y then the value of X and Y a) 3,2 b)2,7 c)1,7 d)4,5
Let the number of non-isomorphic groups of order 10 be X and number of non-isomorphic groups of order 24 be Y then the value of X and Y a) 3,2 b)2,7 c)1,7 d)4,5
answered
Oct 11
in
Mathematical Logic
arun yadav
168
views
0
votes
1
answer
25
Made Easy Test Series:Discrete Math-Mathematical Logic
Consider the following first order logic statement $I)\forall x\forall yP\left ( x,y \right )$ $II)\forall x\exists yP\left ( x,y \right )$ $III)\exists x\exists yP\left ( x,y \right )$ $III)\exists x\forall yP\left ( x,y \right )$ Which one ... true , then $III),IV)$ is true $B)$ If $IV)$ is true , then $II),III)$ is true $C)$ None of these
Consider the following first order logic statement $I)\forall x\forall yP\left ( x,y \right )$ $II)\forall x\exists yP\left ( x,y \right )$ $III)\exists x\exists yP\left ( x,y \right )$ $III)\exists x\forall yP\left ( x,y \right )$ ... $II)$ is true , then $III),IV)$ is true $B)$ If $IV)$ is true , then $II),III)$ is true $C)$ None of these
answered
Oct 11
in
Mathematical Logic
arun yadav
153
views
mathematical-logic
discrete-mathematics
made-easy-test-series
25
votes
4
answers
26
GATE1989-4-i
Provide short answers to the following questions: How many substrings (of all lengths inclusive) can be formed from a character string of length $n$? Assume all characters to be distinct, prove your answer.
Provide short answers to the following questions: How many substrings (of all lengths inclusive) can be formed from a character string of length $n$? Assume all characters to be distinct, prove your answer.
answered
Oct 8
in
Combinatory
varunrajarathnam
2k
views
gate1989
descriptive
combinatory
discrete-mathematics
normal
0
votes
1
answer
27
Ace test series
The formula for the number of positive integers m which are less than p^k and relatively prime to p^k, where p is a prime number and k is a positive integer is__________- A)p^k(p-1) B)(p^(k-2))(p-1) C)p^k(p-2) D)(p^(k-1))(p-1)
The formula for the number of positive integers m which are less than p^k and relatively prime to p^k, where p is a prime number and k is a positive integer is__________- A)p^k(p-1) B)(p^(k-2))(p-1) C)p^k(p-2) D)(p^(k-1))(p-1)
answered
Oct 8
in
Mathematical Logic
Krishnakumar Hatele
53
views
14
votes
2
answers
28
TIFR2013-A-3
Three candidates, Amar, Birendra and Chanchal stand for the local election. Opinion polls are conducted and show that fraction $a$ of the voters prefer Amar to Birendra, fraction $b$ prefer Birendra to Chanchal and fraction $c$ ... $(a, b, c) = (0.49, 0.49, 0.49);$ None of the above.
Three candidates, Amar, Birendra and Chanchal stand for the local election. Opinion polls are conducted and show that fraction $a$ of the voters prefer Amar to Birendra, fraction $b$ prefer Birendra to Chanchal and fraction $c$ ... $(a, b, c) = (0.49, 0.49, 0.49);$ None of the above.
answered
Oct 4
in
Mathematical Logic
Amcodes
1.1k
views
tifr2013
set-theory&algebra
logical-reasoning
29
votes
8
answers
29
GATE2005-IT-33
Let $A$ be a set with $n$ elements. Let $C$ be a collection of distinct subsets of $A$ such that for any two subsets $S_1$ and $S_2$ in $C$, either $S_1 \subset S_2$ or $S_2\subset S_1$. What is the maximum cardinality of C? $n$ $n+1$ $2^{n-1} + 1$ $n!$
Let $A$ be a set with $n$ elements. Let $C$ be a collection of distinct subsets of $A$ such that for any two subsets $S_1$ and $S_2$ in $C$, either $S_1 \subset S_2$ or $S_2\subset S_1$. What is the maximum cardinality of C? $n$ $n+1$ $2^{n-1} + 1$ $n!$
answered
Oct 4
in
Set Theory & Algebra
Shashank Rustagi
4.6k
views
gate2005-it
set-theory&algebra
normal
sets
10
votes
4
answers
30
GATE1998-2.4
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 and Hindi, $11$ persons speak both Hindi and Kannada whereas $13$ persons speak both Kannada and English. How many speak all three languages? $9$ $8$ $7$ $6$
answered
Oct 3
in
Set Theory & Algebra
varunrajarathnam
2.5k
views
gate1998
set-theory&algebra
easy
sets
17
votes
7
answers
31
GATE1996-1.1
Let $A$ and $B$ be sets and let $A^c$ and $B^c$ denote the complements of the sets $A$ and $B$. The set $(A-B) \cup (B-A) \cup (A \cap B)$ is equal to $A \cup B$ $A^c \cup B^c$ $A \cap B$ $A^c \cap B^c$
Let $A$ and $B$ be sets and let $A^c$ and $B^c$ denote the complements of the sets $A$ and $B$. The set $(A-B) \cup (B-A) \cup (A \cap B)$ is equal to $A \cup B$ $A^c \cup B^c$ $A \cap B$ $A^c \cap B^c$
answered
Oct 3
in
Set Theory & Algebra
varunrajarathnam
1.7k
views
gate1996
set-theory&algebra
easy
sets
13
votes
4
answers
32
GATE1995-1.20
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
answered
Oct 3
in
Set Theory & Algebra
varunrajarathnam
5.2k
views
gate1995
set-theory&algebra
normal
sets
21
votes
3
answers
33
GATE1990-3-vi
Which of the following graphs is/are planner?
Which of the following graphs is/are planner?
answered
Oct 2
in
Graph Theory
codeitram
2k
views
gate1989
normal
graph-theory
graph-planarity
descriptive
6
votes
3
answers
34
GATE1988-2xvi
Write the adjacency matrix representation of the graph given in below figure.
Write the adjacency matrix representation of the graph given in below figure.
answered
Oct 2
in
Graph Theory
codeitram
1.2k
views
gate1988
descriptive
graph-theory
graph-connectivity
2
votes
1
answer
35
Generating Function- Where to start?
Hello can anyone suggest good video/book to learn generating functions from?..i tried the nptel lecture..it has some audio lag. and i could not make much out of it..I am well versed in combinatorics but my calculus is weak.. Please suggest some resource that teaches generating functions from scratch
Hello can anyone suggest good video/book to learn generating functions from?..i tried the nptel lecture..it has some audio lag. and i could not make much out of it..I am well versed in combinatorics but my calculus is weak.. Please suggest some resource that teaches generating functions from scratch
answered
Sep 28
in
Combinatory
Himanshu Kumar Gupta
813
views
generating-functions
preparation
0
votes
1
answer
36
Gateforum Test Series: Graph Theory - Graph connectivity
answered
Sep 28
in
Graph Theory
Vineet Pandey
173
views
discrete-mathematics
graph-theory
gateforum-test-series
graph-connectivity
23
votes
6
answers
37
TIFR2018-A-9
How many ways are there to assign colours from range $\left\{1,2,\ldots,r\right\}$ to vertices of the following graph so that adjacent vertices receive distinct colours? $r^{4}$ $r^{4} - 4r^{3}$ $r^{4}-5r^{3}+8r^{2}-4r$ $r^{4}-4r^{3}+9r^{2}-3r$ $r^{4}-5r^{3}+10r^{2}-15r$
How many ways are there to assign colours from range $\left\{1,2,\ldots,r\right\}$ to vertices of the following graph so that adjacent vertices receive distinct colours? $r^{4}$ $r^{4} - 4r^{3}$ $r^{4}-5r^{3}+8r^{2}-4r$ $r^{4}-4r^{3}+9r^{2}-3r$ $r^{4}-5r^{3}+10r^{2}-15r$
answered
Sep 26
in
Graph Theory
manish_pal_sunny
1.8k
views
tifr2018
graph-theory
graph-coloring
25
votes
4
answers
38
GATE2008-IT-3
What is the chromatic number of the following graph? $2$ $3$ $4$ $5$
What is the chromatic number of the following graph? $2$ $3$ $4$ $5$
answered
Sep 26
in
Graph Theory
manish_pal_sunny
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gate2008-it
graph-theory
graph-coloring
normal
28
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7
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39
GATE2003-38
Consider the set \(\{a, b, c\}\) with binary operators \(+\) and \(*\) defined as follows: ... $(x, y)$ that satisfy the equations) is $0$ $1$ $2$ $3$
Consider the set \(\{a, b, c\}\) with binary operators \(+\) and \(*\) defined as follows: ... $(b * x) + (c * y) = c$ The number of solution(s) (i.e., pair(s) $(x, y)$ that satisfy the equations) is $0$ $1$ $2$ $3$
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Sep 25
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Set Theory & Algebra
mayankso
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gate2003
set-theory&algebra
normal
binary-operation
0
votes
3
answers
40
NIELIT 2017 DEC Scientist B - Section B: 43
Which one is the correct translation of the following statement into mathematical logic? “None of my friends are perfect.” $\neg\:\exists\:x(p(x)\land q(x))$ $\exists\:x(\neg\:p(x)\land q(x))$ $\exists\:x(\neg\:p(x)\land\neg\:q(x))$ $\exists\:x(p(x)\land\neg\:q(x))$
Which one is the correct translation of the following statement into mathematical logic? “None of my friends are perfect.” $\neg\:\exists\:x(p(x)\land q(x))$ $\exists\:x(\neg\:p(x)\land q(x))$ $\exists\:x(\neg\:p(x)\land\neg\:q(x))$ $\exists\:x(p(x)\land\neg\:q(x))$
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Sep 22
in
Mathematical Logic
Dhruvil
184
views
nielit2017dec-scientistb
discrete-mathematics
mathematical-logic
first-order-logic
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