GATE Overflow - Recent questions and answers in Mathematical Logic
https://gateoverflow.in/qa/mathematics/discrete-mathematics/mathematical-logic
Powered by Question2AnswerGate 2004| Logics| Let p,q,r, s be 4 primitive statements. Consider these arguments
https://gateoverflow.in/160578/gate-logics-primitive-statements-consider-these-arguments
<p>I tried solving this question using normal approach of writing the whole truth table and evaluating each proposition. This seems to be a very error-prone method. It took over 15-20 minutes. Is there any other method of solving this kind of questions? </p>
<p> </p>
<p><img alt="" src="https://gateoverflow.in/?qa=blob&qa_blobid=15990125441762074211"></p>Mathematical Logichttps://gateoverflow.in/160578/gate-logics-primitive-statements-consider-these-argumentsMon, 16 Oct 2017 16:14:27 +0000Made easy
https://gateoverflow.in/160388/made-easy
<p><img alt="" src="https://gateoverflow.in/?qa=blob&qa_blobid=3453992012750236024"></p>
<p> </p>
<p> </p>
<p>Please tell me how the odd function is getting a positive value over -1/2 to 1/2?</p>Mathematical Logichttps://gateoverflow.in/160388/made-easySun, 15 Oct 2017 17:59:11 +0000Answered: Previous year
https://gateoverflow.in/160006/previous-year?show=160042#a160042
<p>Refer <a rel="nofollow" href="https://gateoverflow.in/41/gate2012_9">https://gateoverflow.in/41/gate2012_9</a> and <a rel="nofollow" href="https://softwareeveryday.wordpress.com/2012/08/07/function-minima-and-maxima-recap/">https://softwareeveryday.wordpress.com/2012/08/07/function-minima-and-maxima-recap/</a> for answer with explanation. </p>Mathematical Logichttps://gateoverflow.in/160006/previous-year?show=160042#a160042Sat, 14 Oct 2017 17:38:47 +0000Answered: logic
https://gateoverflow.in/159631/logic?show=159843#a159843
<p>This is a tautology question in Implication True -> False is only condition on which False we get so in tautology we have to remove it </p>
<p>We can prove it </p>
<p>Case 1:We find for T->F if we get thn not tautology </p>
<p>else tautology</p>
<p>((P=>Q)^(Q=>R))=>(P=>R)</p>
<p>So tautology <img alt="" height="213" src="https://gateoverflow.in/?qa=blob&qa_blobid=10099688835554348055" width="432"></p>
<p> </p>Mathematical Logichttps://gateoverflow.in/159631/logic?show=159843#a159843Sat, 14 Oct 2017 05:20:00 +0000Answered: #Numerical methods-syllabus
https://gateoverflow.in/159600/%23numerical-methods-syllabus?show=159683#a159683
<p>LU decomposition added here in Linear Algebra from Numerical methods which was removed as a whole otherwise.</p>
<p><a rel="nofollow" href="http://gatecse.in/gate-cse-2016-syllabus/">http://gatecse.in/gate-cse-2016-syllabus/</a></p>
<p> </p>Mathematical Logichttps://gateoverflow.in/159600/%23numerical-methods-syllabus?show=159683#a159683Fri, 13 Oct 2017 10:20:54 +0000number of solutions
https://gateoverflow.in/159595/number-of-solutions
When we find rank(A) and rank(AB)<br />
<br />
what are the cases to get<br />
<br />
1.no solution<br />
<br />
2.unique solution<br />
<br />
3.infinite solution<br />
<br />
4.more than one solutionMathematical Logichttps://gateoverflow.in/159595/number-of-solutionsFri, 13 Oct 2017 04:37:00 +0000Answered: UGCNET-Jan2017-III-60
https://gateoverflow.in/159508/ugcnet-jan2017-iii-60?show=159553#a159553
<p>i can write this like:</p>
<p> (R+Q) (P+~Q)</p>
<p> =RP+R~Q+PQ</p>
<p>now option (c)</p>
<p> (R+Q) (P+~Q) (R+~P)</p>
<p>=(RP+R~Q+QP) (R+~P)</p>
<p>=RP+<strong>R~Q</strong>+<em>PQR</em>+<strong>~P~QR</strong>+<em>QP</em></p>
<p>=RP+~QR(1+~P)+PQ(1+R)</p>
<p>=RP+~QR+PQ</p>
<p>SO OPTION(3) WILL BE ANSWER</p>Mathematical Logichttps://gateoverflow.in/159508/ugcnet-jan2017-iii-60?show=159553#a159553Thu, 12 Oct 2017 17:52:33 +0000UGCNET-Jan2017-III-59
https://gateoverflow.in/159507/ugcnet-jan2017-iii-59
<p>Which of the following statements is true ?</p>
<p>(1) The sentence S is a logical consequence of S<sub>1</sub>,…, S<sub>n</sub> if and only if S<sub>1</sub> ∧ S<sub>2</sub> ∧........ ∧ S<sub>n</sub> → S is satisfiable.</p>
<p>(2) The sentence S is a logical consequence of S<sub>1</sub>,…, S<sub>n</sub> if and only if S<sub>1</sub> ∧ S<sub>2</sub> ∧........ ∧ S<sub>n</sub> → S is valid.</p>
<p>(3) The sentence S is a logical consequence of S<sub>1</sub>,…, S<sub>n</sub> if and only if S<sub>1</sub> ∧ S<sub>2</sub> ∧........ ∧ S<sub>n</sub> ∧ ¬S is consistent.</p>
<p>(4) The sentence S is a logical consequence of S<sub>1</sub>,…, S<sub>n</sub> if and only if S<sub>1</sub> ∧ S<sub>2</sub> ∧........ ∧ S<sub>n</sub> ∧ S is inconsistent.</p>Mathematical Logichttps://gateoverflow.in/159507/ugcnet-jan2017-iii-59Thu, 12 Oct 2017 13:31:48 +0000Answered: Math preparation
https://gateoverflow.in/157206/math-preparation?show=159485#a159485
Read Standard books as well as u can refer techtud videos (gate 2018 sprint )Mathematical Logichttps://gateoverflow.in/157206/math-preparation?show=159485#a159485Thu, 12 Oct 2017 12:38:50 +0000Answered: doubt #discrete mathematics
https://gateoverflow.in/157233/doubt-%23discrete-mathematics?show=159484#a159484
Option A<br />
<br />
He is intelligent and tall <br />
<br />
P ^ Q <br />
<br />
but He is neither tall nor intelligent. so<br />
<br />
(P ^ Q)' according to de-morgan's law<br />
<br />
P' V Q'Mathematical Logichttps://gateoverflow.in/157233/doubt-%23discrete-mathematics?show=159484#a159484Thu, 12 Oct 2017 12:36:56 +0000Integration By Trapezoidal Rule (Gate 2014)
https://gateoverflow.in/159428/integration-by-trapezoidal-rule-gate-2014
Using the trapezoidal rule, and dividing the interval of integration into three equal subintervals, the definite integral ∫|x|dx is_______(Upper limit +1& Lower limit -1)Mathematical Logichttps://gateoverflow.in/159428/integration-by-trapezoidal-rule-gate-2014Thu, 12 Oct 2017 09:07:43 +0000Answered: p'q'r is minterm of
https://gateoverflow.in/159039/pq-r-is-minterm-of?show=159206#a159206
${p}'{q}'{r}$ is a minterm then it is equal to 1. If ${p}'{q}'{r}$ = 1 then p=0 , q=0 , and r=1<br />
<br />
Taking options<br />
<br />
A) P v Q = 0<br />
<br />
B) ${p}'$ and R = 0<br />
<br />
C) ${p}'{q}'{r}$ = 1<br />
<br />
D) P and Q and R = 0<br />
<br />
So C is the answerMathematical Logichttps://gateoverflow.in/159039/pq-r-is-minterm-of?show=159206#a159206Wed, 11 Oct 2017 11:22:31 +0000Answered: graphs
https://gateoverflow.in/159198/graphs?show=159200#a159200
<p>[D<sub>n</sub>;/] is poset which is also a distributive lattice and it contains all divisors n.</p>
<p>Here n = 110 which can be written as 2<sup>1 </sup>x 5<sup>1 </sup>x 11<sup>1</sup> = (1+1) * (1+1) * (1+1) = 8 = number of divisors of 110</p>
<p>{1,2,5,10,11,22,55,110} these all are divisors of 110</p>
<p>[D<sub>110 </sub>; /] is also a boolean algebra.</p>Mathematical Logichttps://gateoverflow.in/159198/graphs?show=159200#a159200Wed, 11 Oct 2017 10:58:52 +0000Answered: self_doubt #what is the meaning of this symbole
https://gateoverflow.in/159088/self_doubt-%23what-is-the-meaning-of-this-symbole?show=159188#a159188
We can you write it as (OR)^(NOT) ,i dont think any special name is there for it. THe value of truth table is 1 if only both inputs are 0.Mathematical Logichttps://gateoverflow.in/159088/self_doubt-%23what-is-the-meaning-of-this-symbole?show=159188#a159188Wed, 11 Oct 2017 10:07:05 +0000Answered: propositional logic
https://gateoverflow.in/158918/propositional-logic?show=158940#a158940
$(A)$$((P \rightarrow Q) \rightarrow Q) \\ \equiv ((\sim P \vee Q) \rightarrow Q) \\ \equiv (\sim(\sim P \vee Q) \vee Q) \\ \equiv ((P \wedge \sim Q) \vee Q) \\ \equiv ((P \wedge \sim Q) \vee Q) \\ \equiv (P \vee Q) \wedge (Q \vee \sim Q)) \\ \equiv (P \vee Q) \wedge T \\ \equiv (P \vee Q)$<br />
<br />
$(B)$$(P \leftrightarrow Q) \\ \equiv (P \rightarrow Q) \wedge(Q \rightarrow P) \\ \equiv (\sim P \vee Q) \wedge(\sim Q \vee P) \\ \equiv (\sim P \wedge \sim Q) \vee (P \wedge \sim P) \vee (Q \wedge \sim Q)\vee (Q \wedge P) \\ \equiv (\sim P \wedge \sim Q) \vee F \vee F \vee (Q \wedge P) \\ \equiv (P \wedge Q) (\sim P \wedge \sim Q)$Mathematical Logichttps://gateoverflow.in/158918/propositional-logic?show=158940#a158940Tue, 10 Oct 2017 07:34:07 +0000Answered: number of solutions
https://gateoverflow.in/158719/number-of-solutions?show=158741#a158741
<p>Given , </p>
<p>x + y + z = 17 where x > 1 , y > 2 , z > 3 (or) x >= 2 , y >= 3 , z >= 4.</p>
<p>So we can rewrite the given equation as :</p>
<p> (x + 2) + (y + 3) + (z + 4) = 17 where x,y,z >= 0</p>
<p>==> x + y + z = 8</p>
<p>Now we know :</p>
<blockquote>
<p>Number of non negative integral solutions of x1 + x2 ..........x<sub>n</sub> = r where each of x<sub>i</sub> >= 0 is given by <sup>n-1+r</sup>C<sub>r</sub> </p>
</blockquote>
<p>So here n = 2 , r = 8.</p>
<p><em><strong>So required number of solutions = <sup>3</sup><sup>-1+8</sup>C<sub>8</sub> = <sup>10</sup>C<sub>8</sub></strong></em></p>
<p><em><strong> = <sup>10</sup>C<sub>2</sub></strong></em></p>
<p><em><strong> = 45</strong></em></p>Mathematical Logichttps://gateoverflow.in/158719/number-of-solutions?show=158741#a158741Mon, 09 Oct 2017 15:43:19 +0000rank of matrix
https://gateoverflow.in/158717/rank-of-matrix
A = 1 1 -1<br />
<br />
4 4 -3<br />
<br />
p 2 2<br />
<br />
Rank of matrix A is 2....find PMathematical Logichttps://gateoverflow.in/158717/rank-of-matrixMon, 09 Oct 2017 14:56:44 +0000Gate 2014 ME
https://gateoverflow.in/158633/gate-2014-me
If z is a complex variable, the value of ∫dz/z is____ { Upper limit = 3i & lower limit = 5}<br />
<br />
A. -0.511 - 1.57i<br />
<br />
B. -0.511 + 1.57i<br />
<br />
C. 0.511 - 1.57i<br />
<br />
D. 0.511 + 1.57iMathematical Logichttps://gateoverflow.in/158633/gate-2014-meMon, 09 Oct 2017 11:54:19 +0000Complex Function
https://gateoverflow.in/158630/complex-function
How to make a figure of<br />
<br />
|z - i| = 2.Mathematical Logichttps://gateoverflow.in/158630/complex-functionMon, 09 Oct 2017 11:41:20 +0000matrices
https://gateoverflow.in/158617/matrices
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=7217952334567599098"></p>Mathematical Logichttps://gateoverflow.in/158617/matricesMon, 09 Oct 2017 10:45:55 +0000UGCNET-Jan2017-II-2
https://gateoverflow.in/158414/ugcnet-jan2017-ii-2
<p>Match the following :</p>
<p><strong>List – I List – II </strong></p>
<p>a. Absurd i. Clearly impossible being contrary to some evident truth.</p>
<p>b. Ambiguous ii. Capable of more than one interpretation or meaning.</p>
<p>c. Axiom iii. An assertion that is accepted and used without a proof.</p>
<p>d. Conjecture iv. An opinion preferably based on some experience or wisdom.</p>
<p><strong>Codes : </strong></p>
<p> a b c d</p>
<p>(1) i ii iii iv</p>
<p>(2) i iii iv ii</p>
<p>(3) ii iii iv i</p>
<p>(4) ii i iii iv </p>Mathematical Logichttps://gateoverflow.in/158414/ugcnet-jan2017-ii-2Sun, 08 Oct 2017 20:53:40 +0000#discrete mathematics
https://gateoverflow.in/158370/%23discrete-mathematics
Are Conjunctive Normal Form and Disjunctive Normal Form included in the GATE syllabus?Mathematical Logichttps://gateoverflow.in/158370/%23discrete-mathematicsSun, 08 Oct 2017 15:55:52 +0000mathematics
https://gateoverflow.in/158062/mathematics
the number of integers n with 100<n<200 such that 8 divides (n^2-n-2) and 27 divides (n^2+2n-3)Mathematical Logichttps://gateoverflow.in/158062/mathematicsSat, 07 Oct 2017 10:13:10 +0000Preposition Logic: Verify the solution
https://gateoverflow.in/157896/preposition-logic-verify-the-solution
<p>Which of the following propositional statements is TRUE ?
<br>
<br>
A) ∀x ∀z ∃y [ P(x,y) ]---> ∃y ∀x ∀z [ P(x,y) ]
<br>
<br>
B) ∃y ∀x ∀z [ P(x,y) ]---> ∀x ∀z ∃y [ P(x,y) ]</p>
<p>I think in this</p>
<p>A) False </p>
<p>B) True</p>
<p>Explanation:-</p>
<p><strong>Domain :- Set of all negative numbers. {-inf to -1}</strong></p>
<p>P(x,y) --> x>=y;</p>
<p><strong>A) ∀x ∀z ∃y [ P(x,y) ]---> ∃y ∀x ∀z [ P(x,y) ]</strong></p>
<p><strong>Solving the LHS part </strong>:-</p>
<p>∀x ∀z ∃y [ P(x,y) ]</p>
<p>In this first we have to set the value of x then moving towards the value of y.</p>
<p>x = all value should satisfy the condition</p>
<p>y = x-1;</p>
<p>if x = -3, y = -4 Condition satisfied</p>
<p>if x = -1, y = -2 Condition satisfied.</p>
<p>As we can see for all value of x there exist a value y for which the condition is always holds true. So, LHS part is<strong> True.</strong></p>
<p>Moreover, LHS part always remain true because first we are setting the value of x, and then after setting x we can set y according always. hence it will always true.</p>
<p><strong>Solving the RHS part :-</strong></p>
<p>∃y ∀x ∀z [ P(x,y) ]</p>
<p>In this first we have to set the value of y then moving toward the value of x</p>
<p>y = -1, (Setting the value of y to -1)</p>
<p>x = all value should satisfy the condition</p>
<p>if y = -1, x = -1 Condition satisfied</p>
<p>if y = -1, x = -2 Condition not satisfied,</p>
<p>Since, The condition is not true for all value of x, Hence we can say that RHS part is <strong>false</strong> </p>
<p>Conclusion:-</p>
<p>By the rule of implication, True -> False = False</p>
<p>Hence, part A will be False,</p>
<p><strong>B) ∃y ∀x ∀z [ P(x,y) ]---> ∀x ∀z ∃y [ P(x,y) ]</strong></p>
<p>From the above explanation we can conclude that LHS part of this equation is false and RHS part is True.</p>
<p>By the rule of implication, False -> True = True</p>
<p>Finally conclude that:</p>
<p>A) False</p>
<p>B) True.</p>
<p>Let me know where it is going wrong.</p>Mathematical Logichttps://gateoverflow.in/157896/preposition-logic-verify-the-solutionFri, 06 Oct 2017 15:35:41 +00002008 previous question with answers on propositional logic
https://gateoverflow.in/157475/2008-previous-question-with-answers-on-propositional-logic
Mathematical Logichttps://gateoverflow.in/157475/2008-previous-question-with-answers-on-propositional-logicThu, 05 Oct 2017 09:18:51 +0000Mathematical logic, ace previous papers
https://gateoverflow.in/157474/mathematical-logic-ace-previous-papers
Gate 2008 question on simplificationMathematical Logichttps://gateoverflow.in/157474/mathematical-logic-ace-previous-papersThu, 05 Oct 2017 09:16:28 +0000Ace previous paper on propositional logic
https://gateoverflow.in/157468/ace-previous-paper-on-propositional-logic
Gate 2006 question on binary relationMathematical Logichttps://gateoverflow.in/157468/ace-previous-paper-on-propositional-logicThu, 05 Oct 2017 08:55:31 +0000Is permutations and combination present in discrete math syllabus 2018,,
https://gateoverflow.in/157465/permutations-combination-present-discrete-math-syllabus
Mathematical Logichttps://gateoverflow.in/157465/permutations-combination-present-discrete-math-syllabusThu, 05 Oct 2017 08:35:29 +0000Answered: GATE2004-23, ISRO2007-32
https://gateoverflow.in/1020/gate2004-23-isro2007-32?show=157309#a157309
<p>I have attached two links here. They contain photos clicked from my worked out solution. Hopefully this will clear the doubts arising between option B and D.</p>
<p><a rel="nofollow" href="https://i.stack.imgur.com/zc2j5.jpg">https://i.stack.imgur.com/zc2j5.jpg</a></p>
<p><a rel="nofollow" href="https://i.stack.imgur.com/hgbnR.jpg">https://i.stack.imgur.com/hgbnR.jpg</a></p>Mathematical Logichttps://gateoverflow.in/1020/gate2004-23-isro2007-32?show=157309#a157309Wed, 04 Oct 2017 19:24:12 +0000doubt #discrete mathematics
https://gateoverflow.in/157214/doubt-%23discrete-mathematics
how we solve reoccurrence relation using homogenous equation <br />
<br />
formula ( complete solution = HS + PS )<br />
<br />
#self_doubtMathematical Logichttps://gateoverflow.in/157214/doubt-%23discrete-mathematicsWed, 04 Oct 2017 14:06:43 +0000Answered: doubt
https://gateoverflow.in/157024/doubt?show=157049#a157049
C is a set such that C={a, {a}}<br />
<br />
P(C)= {{ }, {a}, {{a}}, {a,{a}}}<br />
<br />
Now, C Intersection P(C) = {a} which is not empty set, hence above statement is correct.Mathematical Logichttps://gateoverflow.in/157024/doubt?show=157049#a157049Tue, 03 Oct 2017 17:33:28 +0000Answered: discrete mathematics self doubt
https://gateoverflow.in/156991/discrete-mathematics-self-doubt?show=156998#a156998
<p>This is one of the fundamental yet confusing doubts one has in this very topic..</p>
<p>I will try to let u understand through a simple example..</p>
<p><em><strong>Problem : </strong></em><em><strong>How many ways can we select a cricket team of 11 players out of a squad of 13 players.</strong></em></p>
<p>Now here say we have selected a team of 11 players..Hence we are concerned about selection of players only but not arrangement among them..Hence this becomes the problem of combination.</p>
<p><em><strong>Here no of ways = <sup>13</sup>C<sub>11</sub> only.</strong></em></p>
<p>Now lets modify the above problem to make it a permutation problem..</p>
<p><em><strong>Problem : </strong></em><em><strong>How many batting orders can be made from a cricket team of 11 players out of a squad of 13 players.</strong></em></p>
<p>Now here arrangement among the selcted players matter..[ Say Virat Kohli batting at no 4 instead of no 3 , so it will be considered as different only..:) ]</p>
<p>Here no of ways hence = Selection of 11 players from 13 player squad * Batting orders possible among the selected 11players </p>
<p> = <sup>13</sup>C<sub>11</sub> * 11!</p>
<p> = <sup>13</sup>P<sub>11</sub></p>
<p>Hope this helps..:)</p>Mathematical Logichttps://gateoverflow.in/156991/discrete-mathematics-self-doubt?show=156998#a156998Tue, 03 Oct 2017 14:24:28 +0000Answered: Can any one explain this question
https://gateoverflow.in/156771/can-any-one-explain-this-question?show=156815#a156815
it simply means for all x and for all y in the relation R , R(x,y) is same as R(y,x)<br />
<br />
you can also say that R is a symmetric relation .<br />
<br />
consider an example of group theory where elements of the set (1,3,5) on relation R is defined as x is inverse of y .<br />
<br />
now according to the question , if 1 is inverse of 3 , then 3 will be inverse of 1 and so on..<br />
<br />
hope it helps .Mathematical Logichttps://gateoverflow.in/156771/can-any-one-explain-this-question?show=156815#a156815Mon, 02 Oct 2017 17:57:02 +0000Answered: GATE2005-IT-36
https://gateoverflow.in/3783/gate2005-it-36?show=156790#a156790
<p>Answer..<img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=3312955026866523976"></p>Mathematical Logichttps://gateoverflow.in/3783/gate2005-it-36?show=156790#a156790Mon, 02 Oct 2017 16:33:42 +0000Answered: Set theory doubt
https://gateoverflow.in/156646/set-theory-doubt?show=156735#a156735
<p>Given , P and Q are equivalence relations..It is quite possible that the pair {x,y} exists in P and {y,z} exists in Q but {x,z} is not present in either of them..In that case union of P and Q will not contain the pair {x,z} , thereby violating the transitive property..Hence P union Q need not be equivalence relation as for being equivalence relation we need to the relation to be transitive..</p>
<p>For sake of convenience in understanding , we may take P = { (1,1) , (2,2) , (3,3) , (1,3) , (3,1) }</p>
<p> Q = { (1,1) , (2,2) , (3,3) , (3,2) , (2,3) }</p>
<p>So we can see that individually they are equivalence relations but on doing union the pair (1,3) and (3,2) is there due to which (1,2) should also be there..But it wont be included because (1,2) is present in neither P nor Q..</p>
<p>But there wont be such violation in P ∩ Q as if at all (x,y) is present in P and (y,z) is present in Q , then P ∩ Q will not include any of (x,y) or (y,z) and hence the transitivity is preserved.</p>
<p><em><strong>Hence C) option should be correct.</strong></em></p>
<p> </p>
<p><em><strong>NOTE : An empty set cannot generate non empty relation and hence the relation being empty is not reflexive and hence not an equivalence relation..However P and Q are given to be equivalence relations..So u have to take non empty set first and then non empty relation only then we can talk about P U Q or P ∩ Q .</strong></em></p>Mathematical Logichttps://gateoverflow.in/156646/set-theory-doubt?show=156735#a156735Mon, 02 Oct 2017 11:39:37 +0000Asymmetric symmetric asymmetric difference?
https://gateoverflow.in/156657/asymmetric-symmetric-asymmetric-difference
What is meant by antisymmetric , asymmetric ? Any clear difference regarding symmetric, antisymmetric nd asymmetric regarding graph nd example?Mathematical Logichttps://gateoverflow.in/156657/asymmetric-symmetric-asymmetric-differenceMon, 02 Oct 2017 06:21:47 +0000gate previous
https://gateoverflow.in/156568/gate-previous
<p>The given explaination is almost fine.It will be more helpful to me if</p>
<p>someone give other example for this question (specially option d)</p>
<p><a rel="nofollow" href="http://gateoverflow.in/3783/gate2005-it-36">http://gateoverflow.in/3783/gate2005-it-36</a></p>Mathematical Logichttps://gateoverflow.in/156568/gate-previousSun, 01 Oct 2017 16:21:57 +0000Answered: GATE PREV
https://gateoverflow.in/156510/gate-prev?show=156523#a156523
a may be false because ∃x(p(x)∧q(x)) here the vale of x is for both p(x)^q(x)<br />
<br />
for<br />
<br />
∃x(p(x)∧q(x))<br />
<br />
at x=1 => p(1) and q(1)<br />
<br />
at x=9 =>P(9) and q(9)<br />
<br />
.<br />
<br />
.<br />
<br />
.<br />
<br />
.<br />
<br />
at x=n => p(n) and q(n)<br />
<br />
<br />
<br />
<br />
<br />
now for <br />
<br />
(∃xp(x)∧∃xq(x))<br />
<br />
vale of x is distinct for both <br />
<br />
p(1) and q(6)<br />
<br />
p(6) and q (45)<br />
<br />
.<br />
<br />
.<br />
<br />
.<br />
<br />
p(n) and q(m)<br />
<br />
<br />
<br />
<br />
<br />
so by this way it is FALSE...<br />
<br />
<br />
<br />
<br />
<br />
correct me if i am incorrectMathematical Logichttps://gateoverflow.in/156510/gate-prev?show=156523#a156523Sun, 01 Oct 2017 11:24:12 +0000Answered: GATE2015-2_55
https://gateoverflow.in/8259/gate2015-2_55?show=156389#a156389
<p><strong>A)</strong> lets take R(x,y) means x is divisible by y.
<br>
X= {6,8,9}
<br>
Y={2,3}</p>
<p>For all x, there exists any y such that x is divisible by y. It is true in our example.
<br>
but here take y individually, no single y is capable of dividing all x. So it neither implies nor double implies the second one.
<br>
So FALSE.</p>
<p><strong>B)</strong> lets take ...
<br>
R(x,y) means x is divisible by y ,
<br>
S(x,y) means means x/y =2 or 3
<br>
X={4,6,11} Y={2,3}
<br>
(∀x[∃yR(x,y)→S(x,y)])→∀x∃yS(x,y)
<br>
<br>
<em>(Always, make the LHS intentionally true, If we can make RHS false, then it is not tautology)</em>
<br>
<br>
Here for all X, there exists a y such that ..."if x is divisible by y then x/y=2 or 3".... 11 is not divisible by any y...so no problem.
<br>
So LHS is true.
<br>
Go to RHS. For all x, there exists a y such that x/y=2 or 3. Which is false. 11/2 and 11/3 are neither 2 nor 3
<br>
Here T-> F so False. Not tautology.</p>
<p><strong> D) </strong>
<br>
X= 8,12,16
<br>
Y= 2,4
<br>
For all X and for all Y, Y divides X....... It does not imply X divides Y. So FALSE.</p>
<p>
<br>
<br>
We are only left with option C. So it is the ans. The method I follow is to disprove it. Not to prove it. So here also you can try something by taking any example. You cant disprove it. <strong>C is the ans.</strong></p>Mathematical Logichttps://gateoverflow.in/8259/gate2015-2_55?show=156389#a156389Sat, 30 Sep 2017 16:24:44 +0000Answered: GATE1989-14a
https://gateoverflow.in/93179/gate1989-14a?show=156244#a156244
$M(x)\rightarrow x\ is\ mother.\\ Ch(y,x)\rightarrow y\ is\ children\ of\ x\\Lo(x,y)\rightarrow x\ loves\ y.\\ \\$ $$ \forall x\forall y\ [(M(x)\wedge C(y,x))\rightarrow Lo(x,y)]$$Mathematical Logichttps://gateoverflow.in/93179/gate1989-14a?show=156244#a156244Sat, 30 Sep 2017 04:31:52 +0000Answered: Sub-groups
https://gateoverflow.in/152182/sub-groups?show=156157#a156157
<p>Cyclic Group Of <strong>order = 12</strong> with <strong>generator = a</strong> has elements {a,a<sup>2</sup>,a<sup>3</sup>,a<sup>4</sup>,a<sup>5</sup>,a<sup>6</sup>,a<sup>7</sup>,a<sup>8</sup>,a<sup>9</sup>,a<sup>10</sup>,a<sup>11</sup>,a<sup>12</sup>}</p>
<p>Order of generator = Order of group = 12</p>
<p>Order of an element is the smallest positive integer such that a<sup>n</sup> is identity element.</p>
<p>Identity element = a<sup>12</sup></p>
<p>Other Generators are = {a<sup>5</sup>,a<sup>7</sup>,a<sup>11</sup>}</p>
<p>Order of every subgroup divides the order of group.</p>
<table border="1" cellpadding="1" cellspacing="1">
<tbody>
<tr>
<td>Order of Subgroup</td>
<td>SubGroup</td>
</tr>
<tr>
<td>1</td>
<td>a<sup>12</sup></td>
</tr>
<tr>
<td>2</td>
<td>a<sup>6</sup>,a<sup>12</sup></td>
</tr>
<tr>
<td>3</td>
<td>a<sup>4</sup>,a<sup>8</sup>,a<sup>12</sup></td>
</tr>
<tr>
<td>4</td>
<td>a<sup>3</sup>,a<sup>6</sup>,a<sup>9</sup>,a<sup>12</sup></td>
</tr>
<tr>
<td>6</td>
<td>a<sup>2</sup>,a<sup>4</sup>,a<sup>6</sup>,a<sup>8</sup>,a<sup>10</sup>,a<sup>12</sup></td>
</tr>
</tbody>
</table>Mathematical Logichttps://gateoverflow.in/152182/sub-groups?show=156157#a156157Fri, 29 Sep 2017 15:41:09 +0000Discreet maths
https://gateoverflow.in/155926/discreet-maths
I am not able to find out any book for reading path walk close path......in a graph .<br />
<br />
And second group theory<br />
<br />
If anyone know about booksMathematical Logichttps://gateoverflow.in/155926/discreet-mathsThu, 28 Sep 2017 17:44:57 +0000ace test series
https://gateoverflow.in/155704/ace-test-series
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=12892475746197934868">How many edges in graphG=G1 INTERSECTION G2</p>Mathematical Logichttps://gateoverflow.in/155704/ace-test-seriesWed, 27 Sep 2017 17:36:59 +0000X>=y2
https://gateoverflow.in/155563/x-y2
Mathematical Logichttps://gateoverflow.in/155563/x-y2Wed, 27 Sep 2017 08:42:53 +0000GROUP-congruence classes
https://gateoverflow.in/155539/group-congruence-classes
<p>State true or false</p>
<p>1)Let n be an integer,The set Z<sub>n</sub> of congruence classes of integers n with respect to operation of addition is a GROUP.</p>
<p>2)The set Z<sub>n</sub>* of non-zero congruence classes modulo of n, with respect to the operation of multiplication is NOT a GROUP, where n is a prime number.</p>Mathematical Logichttps://gateoverflow.in/155539/group-congruence-classesWed, 27 Sep 2017 06:44:47 +0000Propositional logic
https://gateoverflow.in/155319/propositional-logic
<p><img alt="" height="330" src="http://gateoverflow.in/?qa=blob&qa_blobid=3756445788047149879" width="385"></p>Mathematical Logichttps://gateoverflow.in/155319/propositional-logicTue, 26 Sep 2017 11:34:10 +0000Answered: independent events
https://gateoverflow.in/155050/independent-events?show=155057#a155057
<p>Firstly lets think about it from logical point of view..</p>
<p>If A and B are independent events , so occurence of A does not depend on occurence of B..So probability of occurence of A will remain unaffected irrespective of occurence of B..</p>
<p>Hence P(A | B) meaning P(A) given P(B) is same as P(A) as A is not dependent on B to occur..</p>
<p>Hence for independent events A and B , P(A | B) = P(A)..</p>
<p> </p>
<p>Secondly let us think from mathematical point of view..</p>
<p>We know ,</p>
<p>P(A | B) = P(A ∩ B) / P(B) </p>
<p>And for independent events A and B , P(A ∩ B) = P(A) . P(B)</p>
<p>So ,</p>
<p><em><strong> P(A | B) = P(A) . P(B) / P(B) </strong></em></p>
<p><em><strong> = P(A)</strong></em></p>Mathematical Logichttps://gateoverflow.in/155050/independent-events?show=155057#a155057Mon, 25 Sep 2017 11:47:28 +0000probability
https://gateoverflow.in/155052/probability
You purchase a certain product .The manual states that lifetime T of a product ,defined as the amount of time (in years) the product works properly until it breaks down , satisfies P(T >=t) = e-t/5 , for all t>=0.You purchase and use the product without any prob for two years. The probablility that it breaks down in the third year is ___________________ upto 3 decimal placesMathematical Logichttps://gateoverflow.in/155052/probabilityMon, 25 Sep 2017 11:22:34 +0000Answered: toss and coin
https://gateoverflow.in/154961/toss-and-coin?show=154995#a154995
<p>Here there are 2 cases..Let the probability of head be 'p' as given in question and that of tail be 'q'..Lets find probability for each of the cases and then we sum it up :</p>
<p><em><strong>Case 1 : The toss begins with head :</strong></em></p>
<p>In that case it is possible that first 2 head comes and the game is ended..Else one head , one tail then 2 heads n then game is won and so on..</p>
<p>So P(win in 1st case) = p<sup>2</sup> [First two tosses lead to head] + pqp<sup>2</sup> [First head , then tail , then two heads in a row] + pqpqp<sup>2</sup> + ...</p>
<p> = p<sup>2</sup> [ 1 + pq + (pq)<sup>2</sup> + .......... ]</p>
<p> = p<sup>2</sup> / (1 - pq) [As the internal terms form an infinite G.P with initial term = 1 and common ratio = pq]</p>
<p> </p>
<p><em><strong>Case 2 : Toss begins with tail :</strong></em></p>
<p>So here it is possible that first is tail then two heads and hence game ends and results in win..Else a tail , then a head then a tail then two heads in a row and hence game ends here and so on..</p>
<p>Hence P(win in 2nd case) = qp<sup>2</sup> + qpqp<sup>2</sup> + ..........</p>
<p> = qp<sup>2</sup> [ 1 + qp + (qp)<sup>2</sup> + .......]</p>
<p> = qp<sup>2</sup> / (1 - pq)</p>
<p> </p>
<p>Hence P(win) = P(win in case 1) + P(win in case 2)</p>
<p> = [p<sup>2</sup> / (1 - pq)] + [qp<sup>2</sup> / (1 - pq)]</p>
<p> = p<sup>2 </sup>(1 + q) / (1 - pq)</p>
<p> </p>
<p><em><strong>Hence A) should be the correct answer..</strong></em></p>Mathematical Logichttps://gateoverflow.in/154961/toss-and-coin?show=154995#a154995Mon, 25 Sep 2017 07:33:03 +0000Answered: expectation and variance
https://gateoverflow.in/154963/expectation-and-variance?show=154982#a154982
<p><em><strong>Given E(X) , V(X) , E(Y) , V(Y) , COV(X,Y)</strong></em></p>
<p>As we know :</p>
<blockquote>
<p>Expectation follows linearity of expression (meaning each term can be separated)</p>
</blockquote>
<p>So here : E(X + 2Y) = E(X) + E(2Y)</p>
<p> = E(X) + 2 E(Y) [ Follows from the simple fact : If we multiply each element of Y by 2 , then expectation(mean) is also going to be multiplied by 2 ].</p>
<p> </p>
<p>Now : for Var(X - 2Y + 1) , we need to know :</p>
<blockquote>
<p>Variance only gets affected by scale factor e.g. in -2Y , scale factor is -2 and not by constant terms i.e. '1' here..In other words constant terms do not contribute to variance.</p>
</blockquote>
<p>So general formula for Var(aX + bY + c) = a<sup>2</sup> Var(X) + b<sup>2</sup> Var(Y) + 2ab Cov(X,Y) where Cov(X,Y) is covariance of X and Y[Var(X) is multiplied by a<sup>2</sup> because standard deviation will scale by |a| and hence variance which is square of standard deviation will be squared as well ]</p>
<p> </p>
<p><em><strong>Hence Var(X - 2Y + 1) = 1<sup>2</sup> Var(X) + (-2)<sup>2</sup> Var(Y) + 2(1)(-2) Cov(X,Y)</strong></em></p>
<p><em><strong> = Var(X) + 4 Var(Y) - 4 Cov(X,Y) </strong></em></p>
<p> </p>
<p><em><strong>Cov(X,Y) is found as : E(X.Y) - E(X).E(Y) . Cov(X,X) is nothing but Var(X)..</strong></em></p>Mathematical Logichttps://gateoverflow.in/154963/expectation-and-variance?show=154982#a154982Mon, 25 Sep 2017 07:01:33 +0000