GATE Overflow - Recent questions tagged discrete-mathematics
http://gateoverflow.in/tag/discrete-mathematics
Powered by Question2AnswerCombinatorics : 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!).
<br>
<br>
3. How many solutions of $x1+x2+x3+...+xn=300$ are possible?
<br>
<br>
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[Discrete Maths] predicate Logic
http://gateoverflow.in/133755/discrete-maths-predicate-logic
<pre>
<code>Represent using logic connectives :-
1. "Whenever there is an active alert, all queued messages are transmitted."</code></pre>
<p>Given answer is :-∃x(Alert(x)∧Active(x))→∀y((Message(y)∧Queued(y))→Transmitted(y))</p>
<p>2. Purple mushrooms are poisonous</p>
<p>∀x( purple(x)∧mushroom(x)→ Poisonous(x) )</p>
<p>I need to ask that in first statement why are we using ∃x instead of ∀ in the beginning . I mean it should be same as If there is an active alert then do XYZ.Like in second statement. If i use ∀ in the begriming then it will become "For every object x in universe if x is an active alert then queued messages are transmitted". What is wrong in this?</p>
<p>so ∀(Alert(x) ^ Active(x) -> XYZ).</p>
<p>Why are we using ∃ in the start.?</p>Mathematical Logichttp://gateoverflow.in/133755/discrete-maths-predicate-logicSun, 18 Jun 2017 19:47:44 +0000General Math
http://gateoverflow.in/133745/general-math
Why is $n \leq 2^{h+1} - 1$ equivalent to $h \geq \log_2{\frac{n+1}{2}}$ ? This is applicable to Binary TreesGraph Theoryhttp://gateoverflow.in/133745/general-mathSun, 18 Jun 2017 16:08:26 +0000Set theory
http://gateoverflow.in/133317/set-theory
<p>{ a } ∈ A </p>
<p>but</p>
<p>a ∉ A</p>
<p>why ?</p>
<p>here ' <strong>a is the element of set {a}</strong> ' ,and ' <strong>set {a} is the element of A</strong>" , so " <strong>a also element of A </strong> " . please clear my doubt .</p>Set Theory & Algebrahttp://gateoverflow.in/133317/set-theoryFri, 16 Jun 2017 05:19:09 +0000[Discrete Maths] Group Theory
http://gateoverflow.in/133264/discrete-maths-group-theory
If the group has 40 elements and it has two subgroups namely h and k with order 10 and 20 then what can't be order of (h intersection k)<br />
<br />
a:)10<br />
<br />
b:)20<br />
<br />
c:)3<br />
<br />
d:)1<br />
<br />
e:) 4<br />
<br />
My answer is both c and b and e. Is it correct?Mathematical Logichttp://gateoverflow.in/133264/discrete-maths-group-theoryThu, 15 Jun 2017 15:26:18 +0000[Discrete Maths] Dual of a function
http://gateoverflow.in/133131/discrete-maths-dual-of-a-function
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=16891012436546364258"></p>Mathematical Logichttp://gateoverflow.in/133131/discrete-maths-dual-of-a-functionThu, 15 Jun 2017 04:39:42 +0000[Discrete Maths] First Order Logic
http://gateoverflow.in/133130/discrete-maths-first-order-logic
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=9721698226079299898"></p>Mathematical Logichttp://gateoverflow.in/133130/discrete-maths-first-order-logicThu, 15 Jun 2017 04:38:15 +0000[Discrete Maths] predicate Logic
http://gateoverflow.in/133129/discrete-maths-predicate-logic
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=9721698226079299898"></p>Mathematical Logichttp://gateoverflow.in/133129/discrete-maths-predicate-logicThu, 15 Jun 2017 04:37:42 +0000[Discrete Maths] predicate logic
http://gateoverflow.in/132949/discrete-maths-predicate-logic
<p>Whether following statement is correct?</p>
<blockquote>
<p>Every satisfiable is not tautology.</p>
</blockquote>
<p>I am reading it as:-</p>
<p>1st way :- If it is satisfiable then it is not tautology. So it should be false</p>
<p>2nd way:- It is same as saying that every satisfiable is not tautology because there are some satisfiable which are not tautologies as they are contingencies.</p>
<p>Please clarify which is correct and why?</p>
<p> </p>
<p> </p>Mathematical Logichttp://gateoverflow.in/132949/discrete-maths-predicate-logicWed, 14 Jun 2017 02:08:11 +0000[Discrete maths ] Group theory
http://gateoverflow.in/132942/discrete-maths-group-theory
<p>Let G be abelian, H and K subgroups of G with orders n, m. Then G has subgroup of order lcm(n,m)</p>
<p>I have gone through prrof here:- <a rel="nofollow" href="https://math.stackexchange.com/questions/465742/let-g-be-abelian-h-and-k-subgroups-of-orders-n-m-then-g-has-subgrou">https://math.stackexchange.com/questions/465742/let-g-be-abelian-h-and-k-subgroups-of-orders-n-m-then-g-has-subgrou</a></p>
<p>But this uses assertion :- if G is abelian and n divides |G| then G has a subgroup of order n.How is this assertion true. there are some existing examples where n divides G but there is no subgroup with that order.Please clear with explanation.</p>
<p> </p>Mathematical Logichttp://gateoverflow.in/132942/discrete-maths-group-theoryTue, 13 Jun 2017 22:18:33 +0000first order logic notations
http://gateoverflow.in/132898/first-order-logic-notations
can someone please tell when to interpret this symbol $\Leftrightarrow$ as logical equivalence and when as double implication?Mathematical Logichttp://gateoverflow.in/132898/first-order-logic-notationsTue, 13 Jun 2017 11:18:23 +0000[Discrete Maths] Graph Theory Rosen,Chromatic number
http://gateoverflow.in/132851/discrete-maths-graph-theory-rosen-chromatic-number
<p>What are the chromatic number of following graphs?</p>
<p> </p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=2427432278983743495"></p>
<p>Answer is 6 and 4 respectively.But i am getting 3 for both.</p>
<p>Please someone confirm this?</p>Mathematical Logichttp://gateoverflow.in/132851/discrete-maths-graph-theory-rosen-chromatic-numberTue, 13 Jun 2017 03:38:59 +0000[Discrete Maths] Hamilton Circuit,Rosen p-584,q47
http://gateoverflow.in/132840/discrete-maths-hamilton-circuit-rosen-p-584-q47
<p>How is the following graph is Hamilton?The answer says it has Hamilton circuit</p>
<p>.<img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=8250483095399749125"> </p>Mathematical Logichttp://gateoverflow.in/132840/discrete-maths-hamilton-circuit-rosen-p-584-q47Tue, 13 Jun 2017 02:12:09 +0000Discrete Maths Graph theory
http://gateoverflow.in/132838/discrete-maths-graph-theory
What are the necessary and sufficient conditions for Euler path and Circuit in directed graph?Mathematical Logichttp://gateoverflow.in/132838/discrete-maths-graph-theoryTue, 13 Jun 2017 01:37:24 +0000Discrete maths equivalence relation
http://gateoverflow.in/132834/discrete-maths-equivalence-relation
Consider the equivalence relation<br />
namely, R = (x, y) I x - y is an integer}.<br />
<br />
What is the equivalence class of R={ (x,y) | x-y in integer(Z)} equivalence<br />
relation?<br />
<br />
Given answer is :- (n + 1/2 I n E Z)<br />
<br />
My answer is :- (n*1/2 | n E Z)<br />
<br />
<br />
<br />
Is my answer correct also?Mathematical Logichttp://gateoverflow.in/132834/discrete-maths-equivalence-relationMon, 12 Jun 2017 20:41:16 +0000FREE VARIABLE and VALIDITY related problem
http://gateoverflow.in/132789/free-variable-and-validity-related-problem
<p>Given that </p>
<h2><strong>LHS : </strong><strong>[(∃x,α(x))→β]</strong>
<br>
<strong>RHS</strong> : <strong>[∃x,α(x)→β]</strong></h2>
<p><strong>Here α(x) is a first order formula with x as a free variable, and β is a first order formula with no free variable.</strong></p>
<p>than which one is valid ?</p>
<h3>a) <strong>LHS → RHS</strong></h3>
<h3>b) <strong>RHS → LHS</strong></h3>Mathematical Logichttp://gateoverflow.in/132789/free-variable-and-validity-related-problemMon, 12 Jun 2017 15:05:11 +0000# FREE VARIABLE ( NULL QUANTIFICATION ) Related problem
http://gateoverflow.in/132788/%23-free-variable-null-quantification-related-problem
<h2>Given that </h2>
<h2><strong>LHS : [(∃</strong><strong>x,α(x))→β]</strong></h2>
<h2><strong>RHS : [∃x,α(x)→β]</strong></h2>
<p> </p>
<p>than which one is valid ?</p>
<p>a) <strong>LHS</strong> <strong>→ RHS</strong></p>
<p>b) <strong>RHS → LHS</strong></p>Mathematical Logichttp://gateoverflow.in/132788/%23-free-variable-null-quantification-related-problemMon, 12 Jun 2017 15:00:44 +0000[Discrete Maths] Inclusion/Exclusion Probability
http://gateoverflow.in/132699/discrete-maths-inclusion-exclusion-probability
find the probability that when four numbers from 1 to 100,inclusive,are picked at random with no repetitions allowed,either all are odd,all divisible by 3,or all divisible by 5Mathematical Logichttp://gateoverflow.in/132699/discrete-maths-inclusion-exclusion-probabilitySun, 11 Jun 2017 17:56:46 +0000Series Summation
http://gateoverflow.in/132690/series-summation
Series summation of $S_n$ in closed form?<br />
<br />
$\begin{align*} &S_n = \frac{1}{1.2.3.4} + \frac{1}{2.3.4.5} + \frac{1}{3.4.5.6} + \dots + \frac{1}{n.(n+1).(n+2).(n+3)} \end{align*}$Set Theory & Algebrahttp://gateoverflow.in/132690/series-summationSun, 11 Jun 2017 16:41:53 +0000#Discrete #Combinatorics
http://gateoverflow.in/132665/%23discrete-%23combinatorics
The number of ways in which n distinct objects can be put into two identical boxes so that no box remains empty, is<br />
<br />
a) 2^n - 1<br />
<br />
b) 2^n - 2<br />
<br />
c) 2^(n-1) - 1<br />
<br />
d) None of these<br />
<br />
Please explain your answer.Combinatoryhttp://gateoverflow.in/132665/%23discrete-%23combinatoricsSun, 11 Jun 2017 12:33:15 +0000[Discrete maths] Generating function
http://gateoverflow.in/132616/discrete-maths-generating-function
9. Find the coefficient of x lO in the power series of each of<br />
these functions.<br />
a. (x^2 +x^4 +x^6 +x^8 + .. . )(x^3 +x^6 +x^9 + .. ·)(x^4 +<br />
x^8 +X^12 + ... )<br />
b. (I +x^2+x^4+x^6+ x^8+ ... )(1 +x^4+X^8+X^I2 + ... )<br />
(I +x^6 +x^12 +x^18 + ... )Mathematical Logichttp://gateoverflow.in/132616/discrete-maths-generating-functionSun, 11 Jun 2017 04:26:43 +0000[Discrete maths] Permutations and combinations
http://gateoverflow.in/132599/discrete-maths-permutations-and-combinations
<pre>
How many ways can n books be placed on k distinguishable shelves
a. if no two books are the same ,and the positions of the books on the shelves does not matter?
b. if no two books are the same,and the positions of the books on the shelves matter?
</pre>Mathematical Logichttp://gateoverflow.in/132599/discrete-maths-permutations-and-combinationsSat, 10 Jun 2017 20:38:11 +0000[Discrete Maths] Counting ,Number of solutions to the equation
http://gateoverflow.in/132586/discrete-maths-counting-number-of-solutions-to-the-equation
How many solutions are there to the equation<br />
x1 + x2 + x3 + x4 + x5 = 21,<br />
where xi , i = 1, 2, 3, 4, 5, is a nonnegative integer such that:<br />
<br />
0 ≤ x1 ≤ 3 , 1 ≤ x2 < 4 and 15≤x3 ?Combinatoryhttp://gateoverflow.in/132586/discrete-maths-counting-number-of-solutions-to-the-equationSat, 10 Jun 2017 17:55:46 +0000Mathematical Logic
http://gateoverflow.in/132576/mathematical-logic
<p>What is the difference between =>, <=> and ->?</p>
<p>Are => and -> used in the same way? => is logical implication and <=> is equivalence right? Then why does in some questions, => and <=> is read as 'if then'? '->' symbol is for if then right?</p>
<p>Refer to this: <a rel="nofollow" href="http://gateoverflow.in/3454/gate2007-it-21">http://gateoverflow.in/3454/gate2007-it-21</a></p>Mathematical Logichttp://gateoverflow.in/132576/mathematical-logicSat, 10 Jun 2017 16:22:47 +0000[Discrete maths ] Counting
http://gateoverflow.in/132567/discrete-maths-counting
One hundred tickets numbered from 1 to 100 are sold to 100 different people for a drawing. Four different prizes are to be awarded including the Grand Prize (a trip to Tahiti).<br />
<br />
1) How many ways are there to award the prizes?<br />
<br />
2) How many ways are there to award the prizes if the person holding ticket #47 wins the Grand Prize?<br />
<br />
3) How many ways are there to award the prizes if the person holding ticket #47 wins one of the prizes?<br />
<br />
4) How many ways are there to award the prizes if the person holding ticket #47 does not win a prize?<br />
<br />
I need to clarification on 4th part.I solved it as (Answer for 1- Answer for 3) but it doesn't give the right answer.What did i miss?Mathematical Logichttp://gateoverflow.in/132567/discrete-maths-countingSat, 10 Jun 2017 14:51:30 +0000[Discrete maths] permutations
http://gateoverflow.in/132560/discrete-maths-permutations
How many possibilities are there for the win, place and show (first, second, and third) positions in a horse race with 12 horses if all orders of finish are possible?Mathematical Logichttp://gateoverflow.in/132560/discrete-maths-permutationsSat, 10 Jun 2017 13:52:00 +0000[Discrete Maths] Function,Rosen p-154.1.c
http://gateoverflow.in/132557/discrete-maths-function-rosen-p-154-1-c
<p>Does the following represents function from R->R ?</p>
<p> </p>
<p>f(x)=√x</p>
<p>In ths solution ,it is given:- f(x) is not defined for x < O. </p>
<p>But do we consider it by default as +/-√x or is it only +√x?</p>
<p>I have asked a similar ques:-<a rel="nofollow" href="http://gateoverflow.in/132553/discrete-maths-group-theory">http://gateoverflow.in/132553/discrete-maths-group-theory</a></p>Mathematical Logichttp://gateoverflow.in/132557/discrete-maths-function-rosen-p-154-1-cSat, 10 Jun 2017 13:39:16 +0000[Discrete maths] Group Theory
http://gateoverflow.in/132553/discrete-maths-group-theory
Can the following operation be used with a non empty set to represent a group?<br />
a*b=sqrt($a^{2}+b^{2}$)<br />
<br />
My doubt is,does this operation represents a function or do we include +- sign both as output ?Mathematical Logichttp://gateoverflow.in/132553/discrete-maths-group-theorySat, 10 Jun 2017 13:19:41 +0000[Discrete Maths] Counting
http://gateoverflow.in/132473/discrete-maths-counting
In how many ways can a photographer at a wedding arrange six people in a row, including the bride and groom, if the bride is positioned somewhere to the left of the groom?<br />
<br />
Please suggest non brute force approachMathematical Logichttp://gateoverflow.in/132473/discrete-maths-countingSat, 10 Jun 2017 01:28:04 +0000[Discrete maths] Rosen p-154,functions,7.b
http://gateoverflow.in/132471/discrete-maths-rosen-p-154-functions-7-b
Find range and domain:-<br />
<br />
The function that assigns to each positive integer the number of the digits 0,1,2,3,4,5,6,7,8,9 that do not appear as digits in the decimal representation of the integerMathematical Logichttp://gateoverflow.in/132471/discrete-maths-rosen-p-154-functions-7-bFri, 09 Jun 2017 20:53:22 +0000[Discrete Maths] Predicate Logic,Rosen Ex1.5,problem,9.e
http://gateoverflow.in/132448/discrete-maths-predicate-logic-rosen-ex1-5-problem-9-e
<table>
<tbody>
<tr>
<td>
<p> </p>
</td>
<td>
<p><strong>Q</strong>: Consider the following premises:-</p>
<p>1.1. What is good for corporations is good for the United States.
<br>
2.2. What is good for the United States is good for you.
<br>
3.3. What is good for the corporations is for you to buy lots of stuff.</p>
<p>What are the valid conclusions?</p>
</td>
</tr>
</tbody>
</table>Mathematical Logichttp://gateoverflow.in/132448/discrete-maths-predicate-logic-rosen-ex1-5-problem-9-eFri, 09 Jun 2017 17:50:27 +0000[Discrete Maths] Predicate logic
http://gateoverflow.in/132433/discrete-maths-predicate-logic
Are the following statements same?<br />
<br />
1. Everybody loves exactly one person.<br />
<br />
2.There is exactly one person whom everybody lovesMathematical Logichttp://gateoverflow.in/132433/discrete-maths-predicate-logicFri, 09 Jun 2017 15:21:00 +0000Sheldon Ross chapter 2 questions 26
http://gateoverflow.in/132384/sheldon-ross-chapter-2-questions-26
The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a 2, 3, or 12, the player loses; if the sum is either a 7 or an 11, the player wins. If the outcome is anything else, the player continues to roll the dice until she rolls either the initial outcome or a 7. If the 7 comes first, the player loses, whereas if the initial outcome reoccurs before the 7 appears, the player wins. Compute the probability of a player winning at craps. Hint:Let Ei denote the event that the initial outcome is i and the player wins. The desired probability is $\sum_{i=2}^{12}$P(Ei). To compute P(Ei), define the events Ei,n to be the event that the initial sum is i and the player wins on the nth roll. Argue that P(Ei) =$\sum_{n=1}^{infinte}$ P(Ei,n).Probabilityhttp://gateoverflow.in/132384/sheldon-ross-chapter-2-questions-26Fri, 09 Jun 2017 05:26:43 +0000[Discrete Maths] Rosen,p-46,41.b
http://gateoverflow.in/132366/discrete-maths-rosen-p-46-41-b
Express using predicate,quantifies and connectives:-<br />
<br />
The diagnostic monitor tracks status of all systems except main consoleMathematical Logichttp://gateoverflow.in/132366/discrete-maths-rosen-p-46-41-bFri, 09 Jun 2017 02:36:58 +0000sheldon ross, chapter 2, question 17
http://gateoverflow.in/132359/sheldon-ross-chapter-2-question-17
If 8 rooks are randomly placed on a chessboard, compute the probability that none of the rooks can caputre any of the others. That is compute the probability that no row or file contains more than one rook.Probabilityhttp://gateoverflow.in/132359/sheldon-ross-chapter-2-question-17Fri, 09 Jun 2017 01:32:42 +0000[Discrete Maths] Graph theory
http://gateoverflow.in/132276/discrete-maths-graph-theory
What is the vertex connectivity and edge connectivity of complete graph?<br />
<br />
Is it n or n-1?Graph Theoryhttp://gateoverflow.in/132276/discrete-maths-graph-theoryWed, 07 Jun 2017 22:18:24 +0000Propositional Logic
http://gateoverflow.in/132140/propositional-logic
#Discrete_Maths<br />
#Logic<br />
<br />
I have a doubt in the following concept.<br />
Suppose we're given:<br />
(A v B) ^ ( B v C)...when we open the brackets, do we apply distributive law or just open it..as in<br />
1. A v B ^ B v C = A v B ^ C<br />
2. ((A v B) ^ B) v ((A v B) ^ C) ( by distributive law)<br />
<br />
Which one of the above is correct and why? Also, when exactly do we apply distributive law? I see that sometimes, the brackets are opened just like that and in other cases, distributive law is used. How to differentiate what to do when?<br />
<br />
Please help!Mathematical Logichttp://gateoverflow.in/132140/propositional-logicTue, 06 Jun 2017 23:25:40 +0000first order logic
http://gateoverflow.in/132011/first-order-logic
Is it always the case that implication comes with universal quantifier and conjunction comes with existential quantifier?Mathematical Logichttp://gateoverflow.in/132011/first-order-logicMon, 05 Jun 2017 17:23:35 +0000#general #mathematics #PropositionalLogic How to Choose which one is better answer?
http://gateoverflow.in/131744/%23general-%23mathematics-%23propositionallogic-choose-better
How to Choose which one is a better answer and which is ideally good answer in propositional logic?<br />
<br />
<br />
<br />
let me tell you the context. Let's take an example.<br />
<br />
<br />
<br />
Use quantifiers and predicates with more than one variable to express these statements.<br />
<br />
a) Every computer science student needs a course in Discrete Mathematics.<br />
<br />
now there is various possible solution for this statement but following the same logic.<br />
<br />
solution 1) if the domain is all the people world, S(x) means x is a student of this school, CS(x) x Is in Computer Science class, DM(x) x takes DM course.<br />
<br />
$\forall \left ( S(x)\wedge CS(x)\rightarrow DM(x) \right )$<br />
<br />
solution 2) $\forall \left ( P(x) \right )$<br />
<br />
where P(x), x needs a course in DM and domain consist of all computer science student.<br />
<br />
<br />
<br />
just by changing domain "in words" the scenario changed here. then how to choose the limit of domain and flexibility of it. How to choose which is one is better?Mathematical Logichttp://gateoverflow.in/131744/%23general-%23mathematics-%23propositionallogic-choose-betterSat, 03 Jun 2017 12:00:36 +0000This problem is related to Rule of inference and valid argument .
http://gateoverflow.in/131660/this-problem-is-related-to-rule-inference-and-valid-argument
premises are -<br />
<br />
A<br />
<br />
A → ( B ∨ C )<br />
<br />
B → ¬A<br />
<br />
conclusion -<br />
<br />
C<br />
<br />
<br />
<br />
is valid or not ?Mathematical Logichttp://gateoverflow.in/131660/this-problem-is-related-to-rule-inference-and-valid-argumentFri, 02 Jun 2017 17:59:48 +0000[Discrete Maths] Predicate Logic):-
http://gateoverflow.in/131655/discrete-maths-predicate-logic
A(x) :- Apple on the table.<br />
<br />
Give predicate logic that there is at most one apple on the table.<br />
<br />
1. ∃x∃y(A(x) ^ A(y) ) ->x=y<br />
<br />
2.∀x∀y(A(x) ^ A(y) ) ->x=y<br />
<br />
I know first one is correct,but why second one is not correct?<br />
<br />
If i translate first:- If there exists two apples then they must be same.And if LHS is false,i.e there is no apple then RHS will be true.So it fits both cases of 0 and 1 apple<br />
<br />
If i translate second:- If any two objects in the universe are apples on the table ,then they must be same.It also follows same thing.<br />
<br />
So are both versions correct?Or i am mistaking somehwere?Mathematical Logichttp://gateoverflow.in/131655/discrete-maths-predicate-logicFri, 02 Jun 2017 17:46:33 +0000Kenneth Rosen
http://gateoverflow.in/131575/kenneth-rosen
Show that the following statement is a tautology using Truth Table<br />
<br />
( p ^ q) --> p <br />
<br />
I have some minor doubts in this seemingly simple question.<br />
<br />
First a tautology is a statement which is always True, but while solving the question we get 1 False value, so how is still called a tautology?Mathematical Logichttp://gateoverflow.in/131575/kenneth-rosenFri, 02 Jun 2017 06:13:40 +0000Basic doubt on Modus Tollens Logical Implication
http://gateoverflow.in/131305/basic-doubt-on-modus-tollens-logical-implication
P: Raju Attends Class<br />
Q: Rani Attends Class<br />
<br />
$P \rightarrow Q$ : If Raju Attends the Class, then Rani will attend class.<br />
<br />
For Modus Tollens implication we will consider<br />
($P \rightarrow Q$) $\Lambda$(~Q) $\Leftrightarrow$ ~P<br />
<br />
But why can't we consider<br />
<br />
($P \rightarrow Q$) $\Lambda$(~P) $\Leftrightarrow$ <what could be conclusion here>Mathematical Logichttp://gateoverflow.in/131305/basic-doubt-on-modus-tollens-logical-implicationTue, 30 May 2017 12:03:35 +0000ISBN9780070681880 - Exercise 5.3 - Problem 35 (Kenneth Rosen 7ed SIE)
http://gateoverflow.in/131070/isbn9780070681880-exercise-problem-kenneth-rosen-7ed-sie
How many bit strings contain exactly eight 0s and 10 1s if every 0 must be immediately followed by a 1 ?Mathematical Logichttp://gateoverflow.in/131070/isbn9780070681880-exercise-problem-kenneth-rosen-7ed-sieSat, 27 May 2017 10:20:41 +0000ISBN9780070681880 - Exercise 1.1 - Problem 63 (Kenneth Rosen 7ed SIE)
http://gateoverflow.in/130917/isbn9780070681880-exercise-problem-kenneth-rosen-7ed-sie
A detective has interviewed four witnesses to a crime. From the stories of the witnesses the detective has concluded that if the butler is telling the truth then so is the cook; the cook and the gardener cannot both be telling the truth; the gardener and the handyman are not both lying; and if the handyman is telling the truth then the cook is lying. For each of the four persons can the detective determine whether that person is telling the truth or lying ? Explain your reasoning.Mathematical Logichttp://gateoverflow.in/130917/isbn9780070681880-exercise-problem-kenneth-rosen-7ed-sieFri, 26 May 2017 14:08:20 +0000Counting
http://gateoverflow.in/130747/counting
<p><a rel="nofollow" href="http://gateoverflow.in/?qa=blob&qa_blobid=14775353093279358576">http://gateoverflow.in/?qa=blob&qa_blobid=14775353093279358576</a></p>
<p>I'm unable to understand how the value of K, is equivalent to selecting m integers with repetition allowed, out of n integers (1,2,3,...,n). Kindly explain in detail, preferably visual detail.</p>Mathematical Logichttp://gateoverflow.in/130747/countingThu, 25 May 2017 07:17:34 +0000rosen-prepositional logic - excercise 1.2 16
http://gateoverflow.in/130668/rosen-prepositional-logic-excercise-1-2-16
<p>An explorer is captured by a group of cannibals. There are</p>
<p>two types of cannibals—those who always tell the truth</p>
<p>and those who always lie. The cannibals will barbecue</p>
<p>the explorer unless he can determine whether a particular</p>
<p>cannibal always lies or always tells the truth. He is</p>
<p>allowed to ask the cannibal exactly one question..</p>
<p> </p>
<p><strong>a) </strong>Explain why the question “Are you a liar?” does not</p>
<p>work.</p>
<p><strong>b) </strong>Find a question that the explorer can use to determine</p>
<p>whether the cannibal always lies or always tells the</p>
<p>truth.</p>
<p>in the below link, it mentioned double negation will work. I am not getting what is double negation here. ow the cannibal will consider as two separate question.</p>
<p><a rel="nofollow" href="https://math.stackexchange.com/questions/1078866/is-this-a-correct-solution-to-determining-which-of-two-people-is-the-liar-using">https://math.stackexchange.com/questions/1078866/is-this-a-correct-solution-to-determining-which-of-two-people-is-the-liar-using</a></p>Mathematical Logichttp://gateoverflow.in/130668/rosen-prepositional-logic-excercise-1-2-16Wed, 24 May 2017 11:01:11 +0000Rosen-Prepositional Logic - Excercise 1.2
http://gateoverflow.in/130605/rosen-prepositional-logic-excercise-1-2
<p>You cannot edit a protected Wikipedia entry unless you</p>
<p>are an administrator. Express your answer in terms of <em>e</em>:</p>
<p>“You can edit a protected Wikipedia entry” and <em>a</em>: “You</p>
<p>are an administrator.”
<br>
<br>
the answer given is e----> a</p>
<p>This question seems to be silly. But I am getting so much confused. why can't the answer be a--->e.
<br>
If the user is an Administrator, he can edit the wikipedia entry.
<br>
<br>
please clarify.</p>Mathematical Logichttp://gateoverflow.in/130605/rosen-prepositional-logic-excercise-1-2Wed, 24 May 2017 06:29:11 +0000