GATE Overflow - Recent questions tagged discrete-mathematics
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<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=731647461265627009"></p>Graph Theoryhttp://gateoverflow.in/145083/min-cut-setFri, 18 Aug 2017 03:45:25 +0000probability
http://gateoverflow.in/144372/probability
<p><img alt="" height="201" src="http://gateoverflow.in/?qa=blob&qa_blobid=3799245635594239058" width="652"></p>Probabilityhttp://gateoverflow.in/144372/probabilityTue, 15 Aug 2017 13:28:46 +0000UPSC prelim test
http://gateoverflow.in/144083/upsc-prelim-test
<p><strong>A bag contains 20 balls. 8 balls are green, 7 are white and 5 are red. What is the minimum number of balls that must be picked up from the bag </strong><strong>blind-folded</strong><strong> (without replacing any of it) to be assured of picking </strong><strong>atleast</strong><strong> one ball of each </strong><strong>colour</strong><strong>?</strong></p>
<p>a) 15 </p>
<p>b) 16</p>
<p>c) 17</p>
<p>d)18</p>Combinatoryhttp://gateoverflow.in/144083/upsc-prelim-testMon, 14 Aug 2017 11:16:58 +0000Mathematical logic
http://gateoverflow.in/143835/mathematical-logic
Let P, Q and R be three atomic propositional assertion. Let X denote (PvQ)$\rightarrow$R and Y denote (P$\rightarrow$R)v(Q$\rightarrow$R). Which of the following is a tautology? <br />
<br />
(A) X$equiv$Y<br />
<br />
(B) X$\rightarrow$Y<br />
<br />
(C) Y$\rightarrow$X<br />
<br />
(D) $\dashv$Y$\rightarrow$ $\dashv$X<br />
<br />
ExplainMathematical Logichttp://gateoverflow.in/143835/mathematical-logicSun, 13 Aug 2017 14:32:09 +0000Rosen_Discrete_Mathematics_and_Its_Applications_7th_Edition/Pg.56/Section 1.4 :predicates and quantifiers/Q.43 and Q.44
http://gateoverflow.in/143763/rosen_discrete_mathematics_and_its_applications_7th_edition
43. Determine whether ∀x(P(x)→Q(x)) and ∀xP (x)→∀xQ(x) are logically equivalent. Justify your answer.<br />
<br />
44. Determine whether ∀x(P(x)↔Q(x)) and ∀ x P (x) ↔ ∀ xQ(x) are logically equivalent. Justify your answer.<br />
<br />
<br />
<br />
First I would like to know , how to proceed with such type of questions and what would be the solution..Mathematical Logichttp://gateoverflow.in/143763/rosen_discrete_mathematics_and_its_applications_7th_editionSun, 13 Aug 2017 08:28:08 +0000Mathematical logic
http://gateoverflow.in/143744/mathematical-logic
Consider the logical inference :<br />
<br />
I : if it rains then the cricket match will not be played. <br />
<br />
It did not rain<br />
<br />
Inference :the cricket match was played. <br />
<br />
Is the inference I correct? <br />
<br />
ExplainMathematical Logichttp://gateoverflow.in/143744/mathematical-logicSun, 13 Aug 2017 07:05:17 +0000counting
http://gateoverflow.in/143079/counting
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=13048640937689002281"></p>
<p>what does it mean that "each person has same left and right neighbour "?</p>Combinatoryhttp://gateoverflow.in/143079/countingThu, 10 Aug 2017 16:53:03 +0000General Combinatorics
http://gateoverflow.in/142942/general-combinatorics
<h1>How many 6 digit numbers can be formed from 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6?</h1>Combinatoryhttp://gateoverflow.in/142942/general-combinatoricsThu, 10 Aug 2017 07:34:30 +0000Related to page frame
http://gateoverflow.in/142091/related-to-page-frame
n a 32 bit processor the virtual address is 22 bit:12 bit(Table index : Offset). What is the size of the page frame and the process page table assuming that each entry in the page table is 4 Bytes?Operating Systemhttp://gateoverflow.in/142091/related-to-page-frameSun, 06 Aug 2017 12:44:39 +0000CNF and GNF
http://gateoverflow.in/142000/cnf-and-gnf
<blockquote>
<p><strong>Given answer is (a) but L->AB i think it is wrong because A and B produce something else Previously,</strong></p>
<p><strong> so instead of L->AB there would have given like L->MN </strong></p>
<p><strong> M->c1 and N->S then it was correct . if I am wrong please correct me.</strong></p>
</blockquote>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=14227742477648226448"></p>Compiler Designhttp://gateoverflow.in/142000/cnf-and-gnfSun, 06 Aug 2017 04:56:11 +0000Discrete Maths Kenneth Rosen Nested Quantifiers Ex 1.5 Q8
http://gateoverflow.in/141969/discrete-maths-kenneth-rosen-nested-quantifiers-ex-1-5-q8
<p><em>Let Q(x, y) be the statement “student x has been a contestant
<br>
on quiz show y.” Express each of these sentences
<br>
in terms of Q(x, y), quantifiers, and logical connectives,
<br>
where the domain for x consists of all students at your
<br>
school and for y consists of all quiz shows on television</em>.</p>
<p>
<br>
a) There is a student at your school who has been a contestant
<br>
on a television quiz show.
<br>
b) No student at your school has ever been a contestant
<br>
on a television quiz show.
<br>
c) There is a student at your school who has been a contestant
<br>
on Jeopardy and on Wheel of Fortune.
<br>
d) Every television quiz show has had a student from
<br>
your school as a contestant.
<br>
e) At least two students from your school have been contestants
<br>
on Jeopardy.</p>Mathematical Logichttp://gateoverflow.in/141969/discrete-maths-kenneth-rosen-nested-quantifiers-ex-1-5-q8Sat, 05 Aug 2017 19:07:12 +0000Discrete Mathematics Kenneth H. Rosen Ex 1.4 Predicates and Quantifiers
http://gateoverflow.in/141612/discrete-mathematics-kenneth-rosen-predicates-quantifiers
<p><em>Translate in two ways each of these statements into logical
<br>
expressions using predicates, quantifiers, and logical
<br>
connectives. First, let the domain consist of the students
<br>
in your class and second, let it consist of all people.</em></p>
<p>
<br>
a) Everyone in your class has a cellular phone.
<br>
b) Somebody in your class has seen a foreign movie.
<br>
c) There is a person in your class who cannot swim.
<br>
d) All students in your class can solve quadratic equations.
<br>
e) Some student in your class does not want to be rich</p>Mathematical Logichttp://gateoverflow.in/141612/discrete-mathematics-kenneth-rosen-predicates-quantifiersFri, 04 Aug 2017 11:48:52 +0000Discrete Maths Kenneth H. Rosen Ex 9.4 Relations
http://gateoverflow.in/141221/discrete-maths-kenneth-h-rosen-ex-9-4-relations
<p>Hi, following is a question from Kenneth H. Rosen :</p>
<p>Find the smallest relation containing the relation <strong>R ={(a, b) | a>b}</strong> that is both reﬂexive and symmetric?</p>
<p>The online solution given to the above problem is :</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=14535869392229778214"></p>
<p>I couldn't understand the given solution. Isn't the question asking just for the closure(which should be reflexive and symmetric) of the given relation? Can someone please help me understand this question? </p>Set Theory & Algebrahttp://gateoverflow.in/141221/discrete-maths-kenneth-h-rosen-ex-9-4-relationsWed, 02 Aug 2017 11:36:49 +0000Discrete maths Predicate logic
http://gateoverflow.in/140312/discrete-maths-predicate-logic
<ol>
<li>
<p>( ∃x(P(x))→∃x(Q(x)) ) → ( ∃x(P(x)→Q(x)) )</p>
</li>
<li>
<p>( ∃x(P(x)→Q(x)) ) → ( ∃x(P(x))→∃x(Q(x)) )</p>
</li>
</ol>
<p>Which of the above is valid and for not valid please give me counter example where LHS is true but RHS is false?</p>Mathematical Logichttp://gateoverflow.in/140312/discrete-maths-predicate-logicFri, 28 Jul 2017 09:37:05 +0000counting
http://gateoverflow.in/140204/counting
<p>Let G be a complete undirected graph on 6 vertices. If vertices of G are labeled, then the number of distinct cycles of length 4 in G is equal to</p>
<ol>
<li>15</li>
<li>30</li>
<li>90</li>
<li>360</li>
</ol>Graph Theoryhttp://gateoverflow.in/140204/countingThu, 27 Jul 2017 11:00:45 +0000rosen relations closure ex 35
http://gateoverflow.in/140099/rosen-relations-closure-ex-35
Show that the closure with respect to the property P of<br />
the relation R = {(0, 0), (0, 1), (1, 1), (2, 2)} on the set<br />
{0, 1, 2} does not exist if P is the property<br />
a) “is not reflexive.”<br />
b) “has an odd number of elements.”<br />
<br />
I got the first part, since the relation already is reflexive we can't find a closure for "is not reflexive", but what is meant by closure in the 2nd part?Othershttp://gateoverflow.in/140099/rosen-relations-closure-ex-35Wed, 26 Jul 2017 19:44:01 +0000Rosen: Relations
http://gateoverflow.in/140015/rosen-relations
Draw the Venn Diagram Between Reflexive property, Symmetric property, and Anti-symmetric property?Set Theory & Algebrahttp://gateoverflow.in/140015/rosen-relationsWed, 26 Jul 2017 12:00:12 +0000Tremblay and Manohar
http://gateoverflow.in/139939/tremblay-and-manohar
S={2,a,{3},4} and R={{a},3,4,1} indicate whether the following are true or false<br />
<br />
1) $\varnothing \subset R$<br />
<br />
2)$\varnothing \subseteq \left \{ \left \{ a \right \} \right \}\subseteq R\subseteq E$ E=Universal Set<br />
<br />
3) $\left \{ \varnothing \right \}\subseteq S$<br />
<br />
4) $\varnothing \in R$<br />
<br />
I think 1,2 are true and 3 is false and about 4 nothing is clearSet Theory & Algebrahttp://gateoverflow.in/139939/tremblay-and-manoharWed, 26 Jul 2017 05:57:09 +0000Kenneth Rosen ,DM and its Application Exercise 1.5 Qn 35
http://gateoverflow.in/139839/kenneth-rosen-dm-and-its-application-exercise-1-5-qn-35
Find a common domain for the variables x, y, z,<br />
and w for which the statement $∀x∀y∀z∃w((w \neq x) ∧ (w \neq y) ∧ (w \neq z))$ is true and another common domain<br />
for these variables for which it is false.Mathematical Logichttp://gateoverflow.in/139839/kenneth-rosen-dm-and-its-application-exercise-1-5-qn-35Tue, 25 Jul 2017 15:40:04 +0000Set Theory Doubt
http://gateoverflow.in/139753/set-theory-doubt
Want to verify<br />
<br />
let set $\left | A \right |=n$ and $\left | B \right |=m$<br />
<br />
Then $max(m,n)\leq \left | A\cup B \right |\leq (m+n)$<br />
<br />
$0\leq \left | A\cap B \right |\leq min(m,n)$<br />
<br />
$0\leq \left | A- B \right |\leq \left | n \right |$<br />
<br />
$max(m,n)\leq \left | A\bigoplus B \right |\leq (m+n)$ here $\bigoplus$ is symmetric difference<br />
<br />
$0\leq \left | \overline{A} \right |\leq U$ here $\overline{A}$ is compliment of A and U is universal SetSet Theory & Algebrahttp://gateoverflow.in/139753/set-theory-doubtTue, 25 Jul 2017 07:54:05 +0000Is Summation topic in GATE syllabus
http://gateoverflow.in/139406/is-summation-topic-in-gate-syllabus
Is summation is in Gate 2018 syllabus?Combinatoryhttp://gateoverflow.in/139406/is-summation-topic-in-gate-syllabusSun, 23 Jul 2017 13:37:51 +0000MIT Assignment: Preposition Logic
http://gateoverflow.in/139071/mit-assignment-preposition-logic
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=474484481102173071"></p>
<p>a) x = 1.</p>
<p>b) m is a divisor of n(notation: m|n).</p>
<p>c) n is a prime number.</p>
<p>d) n is a power of a prime.</p>
<p> </p>Mathematical Logichttp://gateoverflow.in/139071/mit-assignment-preposition-logicFri, 21 Jul 2017 17:24:03 +0000Preposition logic "At most" statement
http://gateoverflow.in/139027/preposition-logic-at-most-statement
How to write:-<br />
<br />
"There are at most two apples"<br />
<br />
I think it should be:-<br />
<br />
∀x { A(x) -> ∀y {A(y) -> (x=y) V [ ∀z [A(z) -> ((z=y) V (z=x))]]}}<br />
<br />
is it correct or not?Mathematical Logichttp://gateoverflow.in/139027/preposition-logic-at-most-statementFri, 21 Jul 2017 11:19:00 +0000Permutations
http://gateoverflow.in/138743/permutations
How many different words of 3 length can be formed from a,e,i,o,u containing a,e ?Combinatoryhttp://gateoverflow.in/138743/permutationsWed, 19 Jul 2017 16:28:09 +0000Generating Function Question
http://gateoverflow.in/138235/generating-function-question
<p>I got (x^20).1/(1+x).1/(1-x)^3 .
<br>
Now, if I write this as -
<br>
[x^20].(1+x)^-1.(1-x)^-3</p>
<p>Now, I don't know how to solve this further when we have both factors having negative n.</p>
<p> </p>
<p>Please don't send me here - <a rel="nofollow" href="http://gateoverflow.in/65803/find-number-integral-solutions-using-generating-function">http://gateoverflow.in/65803/find-number-integral-solutions-using-generating-function</a> </p>
<p>As i Didn't get that solution.</p>
<p> </p>
<p>Someone please simplify it.</p>Combinatoryhttp://gateoverflow.in/138235/generating-function-questionSun, 16 Jul 2017 16:32:02 +0000rosen
http://gateoverflow.in/138132/rosen
Show that each of the conditional statement is a tautology<br />
not using truth table<br />
<br />
<br />
<br />
[(p → q) ∧ (q → r)] → (p → r)Mathematical Logichttp://gateoverflow.in/138132/rosenSun, 16 Jul 2017 07:31:06 +0000rosen-eg. 2, section 1.2
http://gateoverflow.in/137983/rosen-eg-2-section-1-2
How can this English sentence be translated into a logical expression?<br />
“You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16<br />
years old.”<br />
<br />
<br />
<br />
The qustion is given above and the answer in the book is:<br />
(r ∧¬s)→¬q<br />
<br />
<br />
<br />
but what if it was like:<br />
<br />
¬q-->(r ∧¬s)<br />
<br />
<br />
<br />
i mean what if you change the left and right hand sides.Mathematical Logichttp://gateoverflow.in/137983/rosen-eg-2-section-1-2Sat, 15 Jul 2017 10:14:46 +0000Probability # Distribution
http://gateoverflow.in/137764/probability-%23-distribution
A traffic office imposes on an average 5 number of penalities daily on traffic violators.Assume that the number of the penalties on different days is independent and follows a Poisson distribution.The probability that there will be less than 4 penalties in a day is _______________ ?Mathematical Logichttp://gateoverflow.in/137764/probability-%23-distributionFri, 14 Jul 2017 09:22:11 +0000Probability
http://gateoverflow.in/137761/probability
A box contains 4 red balls and 6 black balls .Three balls are selected randomly from the box one after another without replacement .The probability that the selected set contains one red ball and two black ball is <br />
<br />
A)1/20<br />
<br />
B) 1/12<br />
<br />
C) 3/10<br />
<br />
D)1/2Mathematical Logichttp://gateoverflow.in/137761/probabilityFri, 14 Jul 2017 09:13:45 +0000Probability
http://gateoverflow.in/137756/probability
A fair (unbiased) coin was tosssed four times in succession and resulted in the following outcomes:<br />
<br />
i)Head , ii) Head, iii) Head iv)Head. The probability of obtaining a "Tail" when a coin is tossed again is<br />
<br />
A) 0<br />
<br />
B) 1/2<br />
<br />
C) 4/5<br />
<br />
D) 1/5Mathematical Logichttp://gateoverflow.in/137756/probabilityFri, 14 Jul 2017 09:08:30 +0000Discrete Maths: First Order Logic - Question in my mind based on question from Kenneth Rosen
http://gateoverflow.in/137425/discrete-maths-first-order-logic-question-question-kenneth
<p>This is not a direct question from Rosen but a question that popped up in my head as I was solving problems from Rosen.</p>
<p>1) ∀<sub>x</sub>∃<sub>y</sub>(x≠y → M(x,y))</p>
<p>2) ∀<sub>x</sub>∃<sub>y</sub>(x≠y ∧ M(x,y))</p>
<p>Here the domain of x,y is the students in your class. M(x,y) represents the statement x has sent y an email.</p>
<p>My question is: Are these two statements equivalent or do they mean different things?</p>
<p>Also, could you please explain the difference between the meaning of these statements (in plain English) if they are different?</p>
<p> </p>
<p> </p>Mathematical Logichttp://gateoverflow.in/137425/discrete-maths-first-order-logic-question-question-kennethWed, 12 Jul 2017 10:50:11 +0000Discrete Maths Rosen 6th Edition - Is the section 'proofs and strategy' part of the Gate syllabus?
http://gateoverflow.in/137347/discrete-maths-rosen-edition-section-proofs-strategy-syllabus
<p>Hi.</p>
<p>I am new to Gate preparation. It would be very helpful if someone could clarify:</p>
<p><em>Are Section 1.6 and 1.7 (Introduction to Proofs and Proof Methods and Strategy) from Kenneth Rosen (6th Edition) part of the Gate Syllabus?</em></p>
<p>I didn't see them mentioned in the syllabus, hence asking.</p>
<p>Thanks.</p>Mathematical Logichttp://gateoverflow.in/137347/discrete-maths-rosen-edition-section-proofs-strategy-syllabusWed, 12 Jul 2017 01:45:17 +0000rosen exercise problem
http://gateoverflow.in/137257/rosen-exercise-problem
<p><strong>How many bit strings of length 10 contain at least three 1s and at least three 0s?</strong></p>
<p><strong>My Approach:-></strong></p>
<p><strong>using product rule</strong></p>
<p><strong> There are 3 subtask following</strong></p>
<p><strong><img alt="T_{1}" src="http://latex.codecogs.com/gif.latex?T_%7B1%7D"> (filling 3 ones in 10 places) =<img alt="\binom{10}{3}" src="http://latex.codecogs.com/gif.latex?%5Cbinom%7B10%7D%7B3%7D"></strong></p>
<p><strong><img alt="T_{2}" src="http://latex.codecogs.com/gif.latex?T_%7B2%7D"> (filling 3 zeros in remaing 7 places) =<img alt="\binom{7}{3}" src="http://latex.codecogs.com/gif.latex?%5Cbinom%7B7%7D%7B3%7D"></strong></p>
<p><img alt="T_{3}" src="http://latex.codecogs.com/gif.latex?T_%7B3%7D"> <strong>(filling remaining 4 places) =</strong> <img alt="2^{4}" src="http://latex.codecogs.com/gif.latex?2%5E%7B4%7D"></p>
<p><strong>So,total number of bit string = <img alt="\binom{10}{3}" src="http://latex.codecogs.com/gif.latex?%5Cbinom%7B10%7D%7B3%7D">*<img alt="\binom{7}{3}" src="http://latex.codecogs.com/gif.latex?%5Cbinom%7B7%7D%7B3%7D">*<img alt="2^{4}" src="http://latex.codecogs.com/gif.latex?2%5E%7B4%7D"> which is greater than <img alt="2^{10}" src="http://latex.codecogs.com/gif.latex?2%5E%7B10%7D">(total number of string).</strong></p>
<p><strong>Now , i want to know what is wrong in my apporach. please explain..</strong></p>Combinatoryhttp://gateoverflow.in/137257/rosen-exercise-problemTue, 11 Jul 2017 12:01:02 +0000Generating Functions Topic Query
http://gateoverflow.in/136893/generating-functions-topic-query
<p>What are the prerequisite to understand Generating Function?</p>
<p>I tried Rosen book, but I didn't get anything from it except the fact that there's some power series involvement in it. </p>
<p>Now, I don't know calculas at all. So, I don't know anything about Power series. </p>
<p>Then I tried watching this video of IIT MADRAS - <a rel="nofollow" href="https://www.youtube.com/watch?v=4d2XEn1j_q4&t=2765s">Lecture 30 - Generating Functions</a> but it has some serious audio/video lag and I'm not able to make much form this video!</p>
<p> </p>
<p>SO, anyone tell me what are things required to understand Generating Function.</p>
<p>P.S - I studied Binomial Theorem Chapter from Class 11th Cbse as binomial expansions are used in generating function.</p>
<p>Please tell me, I'm unable to understand this topic at all.</p>
<p>Thanks</p>Mathematical Logichttp://gateoverflow.in/136893/generating-functions-topic-querySun, 09 Jul 2017 10:53:49 +0000gatebook
http://gateoverflow.in/136871/gatebook
How many bit strings of length $6$ have more no of zeros than ones??<br />
<br />
We can solve this drawing tree but it takes so much time Is there any other way??Combinatoryhttp://gateoverflow.in/136871/gatebookSun, 09 Jul 2017 08:47:05 +0000Rosen
http://gateoverflow.in/136736/rosen
Find the edge chromatic numbers of<br />
a) Cn, where n ≥ 3. (Cycle with n vertices)<br />
<br />
b) Wn, where n ≥ 3 (Wheel with n vertices)<br />
<br />
c)Complete graph with n vertices.Engineering Mathematicshttp://gateoverflow.in/136736/rosenSat, 08 Jul 2017 13:26:57 +0000Rosen Counting Permutations
http://gateoverflow.in/136682/rosen-counting-permutations
I was doing two Questions of rosen counting chapter<br />
<br />
Q1. A group contains n mens and n women. How many ways are there to "arrange" these people in a row if the men and women alternate ?<br />
<br />
Answer is 2(n!)^2, I agree with logic and answer.<br />
<br />
<br />
<br />
Q2. How many ways are there for eight men and five women to stand in a line so that no two women stand next to each other?<br />
<br />
Answer is P(8,8)*P(9,5), I also agree with answer and logic.<br />
<br />
<br />
<br />
My dobut is, The frame of questions are similar but then why can't we solve Q1 with the approach used in Q2 i.e. first permuting n men and then choosing n women places from (n+1) places created my positioning the men and permuting them again i.e P(n,n)*P(n+1,n) ??Combinatoryhttp://gateoverflow.in/136682/rosen-counting-permutationsSat, 08 Jul 2017 07:32:52 +0000Problem related to Equivalence relation
http://gateoverflow.in/136171/problem-related-to-equivalence-relation
<h3><strong>R </strong>is a relation on the set of all <strong>functions </strong>from <strong>Z </strong>to <strong>Z</strong>.</h3>
<h3><strong>R = { (f, g) | for some C ∈ Z , for all x ∈ Z , f(x) - g(x) = C } </strong></h3>
<p>is it <strong>Equivalence relation </strong>or not ?</p>Set Theory & Algebrahttp://gateoverflow.in/136171/problem-related-to-equivalence-relationWed, 05 Jul 2017 13:13:30 +0000Relations
http://gateoverflow.in/136095/relations
Please tell me how to calculate total number of symmetric relations on a set of 5 elements.<br />
<br />
I know the answer but want the proof.Set Theory & Algebrahttp://gateoverflow.in/136095/relationsWed, 05 Jul 2017 04:47:43 +0000relation
http://gateoverflow.in/136077/relation
Plwase explain how inverse of an equivalence relation is an equivalence relation with suitable example?Mathematical Logichttp://gateoverflow.in/136077/relationWed, 05 Jul 2017 00:59:07 +0000Discrete Mathematics for Computer Scientists and Mathematicians , Chapter- 2 , Exercise- 2.2, Question-4
http://gateoverflow.in/135877/discrete-mathematics-computer-scientists-mathematicians
<p>1- Determine the number of 5-combinations of { 1.a , ∞.b ,∞.c , 1.d }. </p>
<p>2<strong>-More generally , develop a formula for the number of r-combinations of a collection of letters a<sub>1 , </sub>a<sub>2 </sub>, ..., a<sub>k </sub>whose repetition numbers are either 1 or ∞.</strong></p>
<p>P.S. i got the answer of 1st but unable to generalize it as asked in 2nd.</p>Combinatoryhttp://gateoverflow.in/135877/discrete-mathematics-computer-scientists-mathematiciansMon, 03 Jul 2017 14:55:09 +0000Discrete Mathematics for Computer Scientists and Mathematicians , Chapter- 2 , Exercise- 2.1, Question-36
http://gateoverflow.in/135833/discrete-mathematics-computer-scientists-mathematicians
Twenty athletes compete in a contest. Each of 3 judges assign 20 different ratings to the 20 athletes. For an athlete to be named winner he must be given the highest rating by at least 2 judges. Compute the fraction of cases for which a winner is named.Combinatoryhttp://gateoverflow.in/135833/discrete-mathematics-computer-scientists-mathematiciansMon, 03 Jul 2017 07:59:32 +0000Equal coefficients
http://gateoverflow.in/135519/equal-coefficients
$\begin{align*} &A = \left ( p x + q \right )^{504} \text{ where p and q are +ve integers and }gcd(p,q) = 1 \\ &\text{Given } \left [ x^4 \right ] = \left [ x^5 \right ] , \text{ where } \left [ x^k \right ] \text{ denote the coefficient of }x^k \text{ in A.} \\ &\text{What is p+q ?} \end{align*}$Combinatoryhttp://gateoverflow.in/135519/equal-coefficientsSat, 01 Jul 2017 04:54:42 +0000#Gate2003
http://gateoverflow.in/135287/%23gate2003
<p>Consider the following logic program P A(x) <- B(x, y), C(y) <- B(x,x) Which of the following first order sentences is equivalent to P?</p>
<p><img alt="" height="81" src="http://www.geeksforgeeks.org/wp-content/uploads/gq/2014/11/GATECS2003Q71.png" width="322"></p>
<p> </p>
<p>Can anyone explain how it can be solved ?</p>Mathematical Logichttp://gateoverflow.in/135287/%23gate2003Thu, 29 Jun 2017 16:09:01 +0000Gate Question for CSE 1995 In Propositional logic
http://gateoverflow.in/135004/gate-question-for-cse-1995-in-propositional-logic
<p>I have an question regarding Propositional Logic,</p>
<p><strong>(! = NOT, V = OR, -> = Implication)</strong></p>
<p>If the proposition <strong>!p -> q</strong> is true, then the truth value of the proposition <strong>!p V (p -> q)</strong> is,</p>
<p>1. true 2. multiple valued, 3. false, 4. cannot be determinded</p>
<p>now from the truth table for <strong>!p V (p->q)</strong> i get <strong>1 1 0 1</strong> which means it is neither true nor false.</p>
<p>so the answer would be either 2 or 4 and the actual answer is 4.</p>
<p>My question is what does 'multiple valued' means in this case? is it just there to misdirect or is there any logic behind it? or am i doing this completely wrong?</p>
<p>Please any kind of advice would appreciated!!!</p>Mathematical Logichttp://gateoverflow.in/135004/gate-question-for-cse-1995-in-propositional-logicTue, 27 Jun 2017 19:51:58 +0000Problem related to irreflexive (Relation)
http://gateoverflow.in/134951/problem-related-to-irreflexive-relation
<h2>Given Relation is <strong>irreflexive</strong> or not ?</h2>
<h1>R = { (a,b) | <strong>there is at least one common link on Web page <span class="marker"><em>a </em></span>and Web page <span class="marker"><em>b</em></span></strong><span class="marker"> </span>}</h1>Set Theory & Algebrahttp://gateoverflow.in/134951/problem-related-to-irreflexive-relationTue, 27 Jun 2017 12:56:38 +0000Sheldon Ross Chapter 1 Example 3c
http://gateoverflow.in/134909/sheldon-ross-chapter-1-example-3c
Ms. Jones has 10 books that she is going to put on her bookshelf. Of these, 4 are mathematics books, 3 are chemistry books, 2 are history books, and 1 is a language book. Ms. Jones wants to arrange her books so that all the books dealing with the same subject are together on the shelf. How many different arrangements are possible? <br />
<br />
Please provide a different explanation rather than the textbook one.Combinatoryhttp://gateoverflow.in/134909/sheldon-ross-chapter-1-example-3cTue, 27 Jun 2017 06:52:14 +0000Combinatorics : Multinomial Coefficients
http://gateoverflow.in/134546/combinatorics-multinomial-coefficients
<p>What's the relationship between combination and polynomial equation? I mean, I am not able to grasp certain points here or let's say connect them into a whole:
<br>
<br>
1. Take a question where it's asked that we have to arrange 10 books : 4 of A, 3 of B, 2 of C, and 1 of D in such a way that each book of similar type remain with its own type so like AAAABBBDCCC, etc. Here we are doing 4!*4!*3!*2!*1!.
<br>
<br>
2. When we take a question of dividing 10 police officers into 3 groups: 10 -> 5,3,2. Here we use 10!/*5!*3!*2!).
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3. How many solutions of $x1+x2+x3+...+xn=300$ are possible?
<br>
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Doubt: What's the difference between 1 & 2? Isn't two equivalent to saying arrange 10 officers like manner of AAAAABBBCC? If we are grouping 10 distinct elements into 3 groups where each group is of same type, then what's the catch here?
<br>
<br>
Second is what's the similarity between 2 and 3? Isn't three equivalent to saying group 300 distinct element into 3 bags/groups?
<br>
<br>
I am totally confuse here and I think I cannot move ahead with my GATE preparation if I cannot clear my doubt on combinatorics which in turn would mean no way of doing probability's tricky questions.</p>
<p> </p>
<p>EDIT: Here's the images of different questions. How do I differentiate between them?</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=9083059680419989013"><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=14165371670004274780"></p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=11854156684812671111"></p>Combinatoryhttp://gateoverflow.in/134546/combinatorics-multinomial-coefficientsSat, 24 Jun 2017 01:58:38 +0000Derangements
http://gateoverflow.in/134403/derangements
How many ways we can put 5 letters L1, L2, L3, L4, L5 in 5 envelopes e1, e2, e3, e4 and e5 (at 1 letter per envelope) so that<br />
<br />
i. no letter is correctly placed?<br />
<br />
ii. at least 1 letter is correctly placed?<br />
<br />
iii. exactly 2 letters are correctly placed?<br />
<br />
iv. at most 1 letter is correctly placed?<br />
<br />
v. at least 1 letter is wrongly placed?<br />
<br />
vi. exactly 1 letter is wrongly placed?Combinatoryhttp://gateoverflow.in/134403/derangementsFri, 23 Jun 2017 03:31:20 +0000[Discrete Maths] Generating functions
http://gateoverflow.in/133773/discrete-maths-generating-functions
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=11849429701953093316"></p>Mathematical Logichttp://gateoverflow.in/133773/discrete-maths-generating-functionsMon, 19 Jun 2017 02:10:59 +0000