GATE Overflow - Recent questions tagged discrete-mathematics
http://gateoverflow.in/tag/discrete-mathematics
Powered by Question2AnswerNull Qunatification Rule
http://gateoverflow.in/130504/null-qunatification-rule
<p>What is the meaning of <strong>NULL Quantification Rule</strong> in Predicate Calculus ?</p>Mathematical Logichttp://gateoverflow.in/130504/null-qunatification-ruleTue, 23 May 2017 03:57:15 +0000Rosen Discreate Probability
http://gateoverflow.in/130478/rosen-discreate-probability
Suppose that one person in 10,000 people has a rare genetic disease. There is an excellent test for the disease; 99.9% of people with the disease test positive and only 0.02% who do not have the disease test positive.<br />
<br />
a) What is the probability that someone who tests positive has the genetic disease?<br />
<br />
b) What is the probability that someone who tests negative does not have the disease?<br />
<br />
My Answers :-<br />
<br />
P(D) = 0.00001 <br />
<br />
P(D') = 0.99999<br />
<br />
P(P/D) = 0.999<br />
<br />
P(P'/D) = 0.001<br />
<br />
P(P/D') = 0.0002<br />
<br />
P(P'/D') = 0.9998<br />
<br />
Answer a) $\frac{(0.999)(0.00001)}{(0.999)(0.00001) + (0.0002)(0.99999)}$<br />
<br />
Answer b) $\frac{(0.9998)(0.99999)}{(0.9998)(0.99999) + (0.001)(0.00001)}$<br />
<br />
Correct me if I am wrong?Probabilityhttp://gateoverflow.in/130478/rosen-discreate-probabilityMon, 22 May 2017 16:34:47 +0000counting
http://gateoverflow.in/130233/counting
Number of words of 4 letters that can be formed with the letters of the word IITJEE is<br />
<br />
a) 42 b) 82 c)102 d) 142Probabilityhttp://gateoverflow.in/130233/countingSat, 20 May 2017 12:04:28 +0000Kenneth rosen chp 6 question 39 counting
http://gateoverflow.in/130005/kenneth-rosen-chp-6-question-39-counting
How many partial functions are there from a set with m elements to a set with n elements, where m and n are positive integers.<br />
<br />
Answer : (n+1)^m <br />
<br />
How . Can anyone please help?Combinatoryhttp://gateoverflow.in/130005/kenneth-rosen-chp-6-question-39-countingWed, 17 May 2017 14:54:04 +0000Kenneth rosen chapter 6 question 41 counting
http://gateoverflow.in/129991/kenneth-rosen-chapter-6-question-41-counting
A palindrome is a string whose reversal is identical to the string. how many bit strings of length n are palindromes ?Combinatoryhttp://gateoverflow.in/129991/kenneth-rosen-chapter-6-question-41-countingWed, 17 May 2017 13:00:29 +0000Kenneth rosen chapter 6 question 17 counting
http://gateoverflow.in/129976/kenneth-rosen-chapter-6-question-17-counting
How many strings of 5 ASCII characters contain the character @ atleast once ? [ NOTE : there are 128 ascii characters ] <br />
<br />
<br />
<br />
Answer is : 1,321,368,961<br />
<br />
Can anyone explain how ?Combinatoryhttp://gateoverflow.in/129976/kenneth-rosen-chapter-6-question-17-countingWed, 17 May 2017 11:15:26 +0000Rosen Doubt
http://gateoverflow.in/129359/rosen-doubt
<p>Express the specification “<strong>The automated reply cannot be sent when the file system is full”</strong></p>
<p>using logical connectives.</p>
<p> </p>Mathematical Logichttp://gateoverflow.in/129359/rosen-doubtFri, 12 May 2017 08:20:38 +0000Graph Theory
http://gateoverflow.in/129355/graph-theory
algorithm to find more than one path between any two vertices of a graph G=(V,E) , with a complexity of O(VE) ?Graph Theoryhttp://gateoverflow.in/129355/graph-theoryFri, 12 May 2017 08:04:59 +0000Graph theory and Applications Bondy and Murty Exercise Qn 1.9
http://gateoverflow.in/129036/graph-theory-and-applications-bondy-and-murty-exercise-qn-1
<p>A k partite graph is one where vertex set can be partitioned into k subsets so that no edge has both end in any one subset.</p>
<p>A complete k partite graph is one that is simple and in which each vertex is joined to every other vertex that is not in the same subset. The complete m-partite graph on n vertices in which each part has either floor(n/m) or ceil(n/m) vertices is denoted by T<sub>m,n</sub> . Show that</p>
<p>a) | E(T<sub>m,n</sub>) | = $\binom{n-k}{2} + (m-1)\binom{k+1}{2} , k = \left \lfloor n/m \right \rfloor$</p>
<p>b) If G is a complete m-partite graph on n vertices then | E(G) | <= | E(T<sub>m,n</sub>)|, with equality only if G isomorphic to T<sub>m,n</sub></p>Graph Theoryhttp://gateoverflow.in/129036/graph-theory-and-applications-bondy-and-murty-exercise-qn-1Tue, 09 May 2017 16:09:49 +0000#rosen , #counting , #5.1 , 41(a)
http://gateoverflow.in/128608/%23rosen-%23counting-%235-1-41-a
in how many ways can a photographer at a wedding arrange 6 people in a row from a group of 10 people, where the bride and the groom are among these 10 people , if<br />
<br />
a) the bride must next to the groom<br />
<br />
my solution is since given that bride and groom must be next to each other then now we have to select 4 person from 8 that is 8c4 now we have to arrange these people since bride and groom next to each other then take BG as one block and rest four __ , __ , __ , __ ,<br />
<br />
now arranging BG , __ , __ ,__, __ will take 5! , and arranging BG mutually is 2! ,<br />
<br />
so total ways is 8c4*5!*2!<br />
<br />
but answer is given 240 (rosen )Combinatoryhttp://gateoverflow.in/128608/%23rosen-%23counting-%235-1-41-aSun, 07 May 2017 17:51:30 +0000keneth r rosen
http://gateoverflow.in/128476/keneth-r-rosen
<p>HOW TO SOLVE THIS USING BOOLEAN ALGEBRA; <img alt="" height="71" src="http://gateoverflow.in/?qa=blob&qa_blobid=986322319398299267" width="199"></p>Mathematical Logichttp://gateoverflow.in/128476/keneth-r-rosenSun, 07 May 2017 13:40:59 +0000keneth r rosen
http://gateoverflow.in/128471/keneth-r-rosen
how to prove that sum of all the vertices in a graph G is equal to twice the number of edges in G.<br />
<br />
please explain step by step .Graph Theoryhttp://gateoverflow.in/128471/keneth-r-rosenSun, 07 May 2017 13:28:40 +0000keneth r rosen
http://gateoverflow.in/128470/keneth-r-rosen
<p>how to prove that graph G with<strong> </strong>e= v - 1 that has no circuit is a tree.</p>Graph Theoryhttp://gateoverflow.in/128470/keneth-r-rosenSun, 07 May 2017 13:26:36 +0000keneth r rosen
http://gateoverflow.in/128469/keneth-r-rosen
If a,b be elements of a Boolean algebra then how to show that (a∨b)' = a' ∧ b'.<br />
<br />
please explain step by step in easiest way possibleMathematical Logichttp://gateoverflow.in/128469/keneth-r-rosenSun, 07 May 2017 13:23:32 +0000how to remove ambiguity that whether should take lower or upper shield in counting problem
http://gateoverflow.in/128445/remove-ambiguity-whether-should-lower-shield-counting-problem
how many positive integer between 50 and 100 ,<br />
<br />
(a) divisible by 7<br />
(b) divisible by 11<br />
(c) divisible by 7 and 11 ?<br />
<br />
people generally answer this question as<br />
<br />
(a) lowershield [((100-50)-1 )/7] = 7 , yess its true the no is 56 , 63 , 70, 84, 91 , 98 <br />
(b) lowershield [((100-50)-1 )/11] = 4 , oops its not true the no is 55,66,77,88,99<br />
<br />
(c) lowershield [((100-50)-1 )/lcm(7,11)] = 0 , oops its not true the no is 77<br />
<br />
if u will say to take uppershield then lets come<br />
<br />
(a) uppershield[((100-50)-1)/7]=7 , yess its true<br />
<br />
(b) uppershield[((100-50)-1)/11]=5 , yess its true<br />
<br />
(c) uppershield[((100-50)-1)/lcm(7,11)]=1 , yess its true<br />
<br />
now come to another question<br />
<br />
how many no is divisible between 5 to 31 is divisible by 4 ?<br />
<br />
case1 : loweshield [((31-5)-1)/4]=6 , yess its true 8,12,16,20,24,28<br />
<br />
case 2 : uppershierld [((31-5)-1/4)]=7 , oops its wrong<br />
<br />
how can we remove this ambiguity ?<br />
<br />
if u will argue that check the no ? but we have to check for large no like 50 to 300000 then how can u count ?<br />
<br />
my doubt is , is any way to remove this ambiguity ?Mathematical Logichttp://gateoverflow.in/128445/remove-ambiguity-whether-should-lower-shield-counting-problemSun, 07 May 2017 09:54:44 +0000Kenneth Rosen Ch 6 Counting
http://gateoverflow.in/128256/kenneth-rosen-ch-6-counting
#discrete<br />
<br />
6 runners are in a 100 yard dash. Find the ways possible for 3 medals to be awarded if ties are possible.Mathematical Logichttp://gateoverflow.in/128256/kenneth-rosen-ch-6-countingFri, 05 May 2017 17:44:44 +0000Rosen Mathematical Logic Example Simple question with "Only if"
http://gateoverflow.in/127998/rosen-mathematical-logic-example-simple-question-with-only
<p>You can access the internet <strong>only if</strong> you are a computer science major or you are not a freshman.
<br>
<br>
You can access the internet (p)
<br>
you are a computer science major (q)
<br>
you are a freshman (r)
<br>
<br>
Why the ans is <strong>p-> (q v r ) </strong>?
<br>
Why not <strong>(q v r ) -> p </strong> ?</p>Mathematical Logichttp://gateoverflow.in/127998/rosen-mathematical-logic-example-simple-question-with-onlyWed, 03 May 2017 20:34:16 +0000PGEE 2017
http://gateoverflow.in/127522/pgee-2017
<p>P1: Number of one-to-one function from A$\bigotimes A$ is 1.</p>
<p>P2: Function A->B is bijective C->D is bijective then A<sup>C</sup> and B<sup>D</sup> is also bijective.</p>
<p>P3: Consider set A,B,C,D then ($A\cup B\cap C\cup D) == (A\cap B\cup C\cap D$)</p>
<p> </p>
<p>which of above statement are true and which are false</p>Set Theory & Algebrahttp://gateoverflow.in/127522/pgee-2017Sun, 30 Apr 2017 18:16:11 +0000combinatorics
http://gateoverflow.in/127196/combinatorics
In how many ways 8 different shirts can be distributed to 4 different people so that each will get 2 shirts?Combinatoryhttp://gateoverflow.in/127196/combinatoricsFri, 28 Apr 2017 09:40:08 +0000combinatorics
http://gateoverflow.in/127195/combinatorics
Number of 5 digit number having there digits in non decreasing order (from left to right) constructed by using the digits belonging to the set {1, 2, 3, 4, 5, 6, 7, 8, 9} ?Combinatoryhttp://gateoverflow.in/127195/combinatoricsFri, 28 Apr 2017 09:38:19 +0000Generalised permutation combinations
http://gateoverflow.in/126603/generalised-permutation-combinations
In how many ways can a dozen books be placed on four distinguishable shelves<br />
<br />
if no two books are the same, and the positions of the books on the shelves matter?<br />
<br />
(Hint: Break this into 12 tasks, placing each book separately. Start with the sequence 1,2,3,4 to<br />
<br />
represent the shelves. Represent the books by bi, i = 1, 2, ..., 12. Place b1 to the right of one of<br />
<br />
the terms in 1, 2, 3, 4. Then successively place b2, b3, ..., and b12.)Combinatoryhttp://gateoverflow.in/126603/generalised-permutation-combinationsSat, 22 Apr 2017 21:58:10 +0000rosen
http://gateoverflow.in/126428/rosen
<p><big>A shelf holds 12 books in a row. How many ways are there to choose five books so that no two adjacent books are chosen?</big></p>Combinatoryhttp://gateoverflow.in/126428/rosenFri, 21 Apr 2017 15:46:38 +0000Combinatorics
http://gateoverflow.in/126365/combinatorics
A bagel shop has onion bagels, poppy seed bagels, egg bagels, salty bagels, pumpernickel bagels, sesame seed bagels, raisin bagels, and plain bagels. <br />
<br />
How many ways are there to choose<br />
<br />
a dozen bagels with at least three egg bagels and no more than two salty bagels?Combinatoryhttp://gateoverflow.in/126365/combinatoricsFri, 21 Apr 2017 00:26:11 +0000rosen discrete
http://gateoverflow.in/126217/rosen-discrete
How many strings of six lowercase letters of the<br />
English alphabet contain exactly two vowel?Combinatoryhttp://gateoverflow.in/126217/rosen-discreteWed, 19 Apr 2017 18:04:50 +0000Kenneth Rosen- Mathematical logic
http://gateoverflow.in/126106/kenneth-rosen-mathematical-logic
Express below using quantifiers :<br />
<br />
"At least one mail message among the non-empty set of messages, can be saved if there is a disk with more than 10 kilobytes of free space"<br />
<br />
Answer was given as :<br />
<br />
(∃x F(x,10))→∃x S(x)<br />
<br />
Where F(x,y) Disk x has more than y kilobytes of free space<br />
<br />
S(x) Mail message x can be saved.<br />
<br />
Can somebody explain me the solution?Mathematical Logichttp://gateoverflow.in/126106/kenneth-rosen-mathematical-logicTue, 18 Apr 2017 18:32:43 +0000Kenneth Rosen - Mathematical logic
http://gateoverflow.in/126105/kenneth-rosen-mathematical-logic
Let T(x,y) mean that student x likes cuisine y, where the domain for x consists of all students at your school and the domain y consists of all cuisines.<br />
<br />
What is meant by the below expression?<br />
<br />
∀x∀z∃y ((x≠z)→ ∼(T(x,y) ^ T(z,y)))Mathematical Logichttp://gateoverflow.in/126105/kenneth-rosen-mathematical-logicTue, 18 Apr 2017 18:27:22 +0000Xerox Note Of Discrete Math
http://gateoverflow.in/125916/xerox-note-of-discrete-math
Are the following implications right or wrong and how?<br />
<br />
1. for_every_x( p(x)-->q(x) ) ---> (for_every_x(p(x)) --> for_every_x(q(x)))<br />
<br />
2. (der_exist_x(p(x))-->der_exist_x(q(x))) ---> der_exist_x (p(x)-->q(x))Study Resourceshttp://gateoverflow.in/125916/xerox-note-of-discrete-mathMon, 17 Apr 2017 12:10:31 +0000Rosen chapter-6 (counting)
http://gateoverflow.in/125539/rosen-chapter-6-counting
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: 0$\leq$ x1$\leq$10 ?Combinatoryhttp://gateoverflow.in/125539/rosen-chapter-6-countingFri, 14 Apr 2017 16:31:03 +0000kenneith rosen
http://gateoverflow.in/125412/kenneith-rosen
At least how many numbers should be selected from the set {1, 5, 9, 13, …125} to be assured that two of the numbers selected have a sum of 146?Set Theory & Algebrahttp://gateoverflow.in/125412/kenneith-rosenThu, 13 Apr 2017 14:32:59 +0000graph theory
http://gateoverflow.in/124669/graph-theory
A graph consists of only one vertex,which is isolated ..Is that graph<br />
<br />
A) a complete graph ???<br />
<br />
B) a clique???<br />
<br />
C) connected graph ???<br />
<br />
Please explain your answer ...Graph Theoryhttp://gateoverflow.in/124669/graph-theoryFri, 07 Apr 2017 17:38:26 +0000First Order Logic
http://gateoverflow.in/124596/first-order-logic
Can some explain the basics of First Order Logic like what various terms means, what are various properties ?Study Resourceshttp://gateoverflow.in/124596/first-order-logicFri, 07 Apr 2017 12:14:08 +0000logic
http://gateoverflow.in/124485/logic
<p>can somebody explain the intution behind this ...? i am not able to get ....and i dont want to by-heart ....<img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=16599048480255075131"></p>Mathematical Logichttp://gateoverflow.in/124485/logicThu, 06 Apr 2017 19:49:49 +0000UGC NET 2014 (JUNE)
http://gateoverflow.in/124416/ugc-net-2014-june
How many different truth tables of the<br />
compound propositions are there that<br />
involve the propositions p & q ?<br />
(A) 2 (B) 4<br />
(C) 8 (D) 16Mathematical Logichttp://gateoverflow.in/124416/ugc-net-2014-juneThu, 06 Apr 2017 13:18:12 +0000ISI 2004 MIII
http://gateoverflow.in/124234/isi-2004-miii
<p>Let <em>X</em> be an nonempty set and let <em>P(X) </em>denote the collection of all subset of <em>X</em>. Define $\textit{f}:\textit{X*P(X)}\rightarrow \mathbb{R}$ by</p>
<p>f(x,A) = $\left\{\begin{matrix} 1 if x \epsilon A & \\ 0 if x\not\epsilon A & \end{matrix}\right.$</p>
<p>Then $f\left ( x,A\cup B \right )$ equals</p>
<p>A) f(x,A)+f(x,B)</p>
<p>B) f(x,A)+f(x,B)-1</p>
<p>C) f(x,A)+f(x,B) - f(x,A).f(x,B).</p>
<p>D) f(x,A)+ $\left | f(x,A)-f(x,B) \right |$</p>Mathematical Logichttp://gateoverflow.in/124234/isi-2004-miiiWed, 05 Apr 2017 15:27:34 +0000combinatorics
http://gateoverflow.in/124223/combinatorics
THERE ARE 10 PRIZES AND 4 STUDENTS. WE WANT ONLY TWO STUDENTS TO GET THE PRIZES. HOW MANY WAYS ARE THERE??Mathematical Logichttp://gateoverflow.in/124223/combinatoricsWed, 05 Apr 2017 14:06:06 +0000Set theory
http://gateoverflow.in/123552/set-theory
Justify your answer with valid reason<br />
<br />
$\phi \epsilon \left \{ A \right \} \wedge \phi \subseteq \left \{ A \right \}$<br />
<br />
Where A is an non empty set $\left | A \right | \neq 0$Algorithmshttp://gateoverflow.in/123552/set-theorySun, 02 Apr 2017 16:03:25 +0000Tremblay Manohar
http://gateoverflow.in/123550/tremblay-manohar
Write the set<br />
<br />
$\left \{ \phi ,\left \{ \phi \right \} \right \}-\phi$Algorithmshttp://gateoverflow.in/123550/tremblay-manoharSun, 02 Apr 2017 15:56:37 +0000Tremblay Manohar
http://gateoverflow.in/123549/tremblay-manohar
Write the set<br />
<br />
$\phi \cap \left \{ \phi \right \}$Algorithmshttp://gateoverflow.in/123549/tremblay-manoharSun, 02 Apr 2017 15:55:01 +0000Set theory
http://gateoverflow.in/123547/set-theory
Let S be a set of n elements and let p(S) be its power set. Then find total number of ordered pairs such that $s1 \cap s2 = \phi$ where s1 & s2 are subset of p(s)Algorithmshttp://gateoverflow.in/123547/set-theorySun, 02 Apr 2017 15:43:22 +0000Rosen Logic Exercise.
http://gateoverflow.in/123456/rosen-logic-exercise
<p>Need explanation for the answer.
<br>
<img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=4621047533555316713"></p>Mathematical Logichttp://gateoverflow.in/123456/rosen-logic-exerciseSun, 02 Apr 2017 08:56:33 +0000logic
http://gateoverflow.in/123124/logic
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=3341390517395662876"></p>
<p><strong>Is there anyother way to solve this problem other than drawing the complete truth table ?</strong></p>Mathematical Logichttp://gateoverflow.in/123124/logicFri, 31 Mar 2017 17:23:01 +0000group theory
http://gateoverflow.in/122536/group-theory
<p><em><strong>"the union of two sub-groups neednot be a sub-group"</strong></em>.can some-body prove without using counter example ...</p>Set Theory & Algebrahttp://gateoverflow.in/122536/group-theorySat, 25 Mar 2017 16:50:42 +0000group theory
http://gateoverflow.in/122528/group-theory
<p><em><strong>"The necessary and sufficient condition for a subset of a group to be called as sub-group is it should satisfy the algebraic structure property".</strong></em></p>
<p>can somebody tell why this should be TRUE. We are concluding it as sub-group just by seeing only algebraic structure property satisfying and not checking for identity element,inverse ....why ?</p>Set Theory & Algebrahttp://gateoverflow.in/122528/group-theorySat, 25 Mar 2017 14:43:56 +0000group theory
http://gateoverflow.in/122494/group-theory
<p>which among the following statements is TRUE ?</p>
<p><strong><em>S1 :</em> ( { 0,1,2....(m-1) } , +<sub>m</sub> )</strong> where<em><strong> +<sub>m</sub></strong></em> stands for "addition-modulo-m"</p>
<p><strong><em>S2 :</em> ( {0,1,2....m} , +<sub>m</sub> )</strong> where <em><strong>+<sub>m</sub></strong></em> stands for "addition-modulo-m".</p>
<p>A) ONLY S1 is a group.</p>
<p>B) ONLY S2 is a group.</p>
<p>C) BOTH S1 AND S2 are groups.</p>
<p>D) NEITHER S1 NOT S2 is a group.</p>
<p> </p>Set Theory & Algebrahttp://gateoverflow.in/122494/group-theorySat, 25 Mar 2017 09:31:51 +0000group theory
http://gateoverflow.in/122470/group-theory
<p> Is <strong>( {0},* )</strong> a group ? ( where * stands for multiplication operation ).</p>
<p>Please provide explanations with your answer </p>Set Theory & Algebrahttp://gateoverflow.in/122470/group-theoryFri, 24 Mar 2017 21:42:49 +0000set theory
http://gateoverflow.in/122391/set-theory
<p>Which of the following statements about the POSET diagram given below is TRUE ?</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=14828831978029863392"></p>
<p>For a lattice with 8 elements to be called as boolean algebra </p>
<p>A) It is a necessary and a sufficient condition that it should be isomorphic to above POSET diagram.</p>
<p>B) It is necessary condition to be isomorphic to above POSET diagram but not a sufficient condition. </p>
<p>C) It is sufficient condition but not a necessary condition.</p>
<p>Which among A),B),C) is true ..? why ? please explain ?</p>Set Theory & Algebrahttp://gateoverflow.in/122391/set-theoryThu, 23 Mar 2017 20:45:56 +0000set theory
http://gateoverflow.in/122378/set-theory
<p>STATE TRUE OF FALSE</p>
<p>The Poset<strong> [ D<sub>n</sub>,/ ] </strong>is a distributive lattice for any positive integer n,<em> where D<sub>n</sub> stands for divisors of n and relation "/" is a divides operation (i.e) (a,b) belongs to relation,R iff a divides b</em>.</p>
<p>Please support your answer by giving explanations.</p>Set Theory & Algebrahttp://gateoverflow.in/122378/set-theoryThu, 23 Mar 2017 17:01:49 +0000set theory
http://gateoverflow.in/122325/set-theory
How many among A),B),C),D) are distributive lattices ?<br />
<br />
A) [ P(A),$\subseteq$] where A is a set with n elements.<br />
<br />
B) [ P(A),$\subset$] where A is a set with n elements.<br />
<br />
C) [ P(A),$\supset$] where A is a set with n elements.<br />
<br />
D) [ P(A),$\supseteq$] where A is a set with n elements.Set Theory & Algebrahttp://gateoverflow.in/122325/set-theoryWed, 22 Mar 2017 17:23:25 +0000set theory
http://gateoverflow.in/122322/set-theory
<p><em><strong>"Is every finite lattice a bounded lattice ??"</strong></em> please support your answer by giving some informal proof ?</p>Set Theory & Algebrahttp://gateoverflow.in/122322/set-theoryWed, 22 Mar 2017 16:34:34 +0000set theory
http://gateoverflow.in/122314/set-theory
<p><em><strong>"Every sub-lattice of a distributive lattice is distributive"</strong></em>. can somebody prove it ??</p>
<p> </p>Set Theory & Algebrahttp://gateoverflow.in/122314/set-theoryWed, 22 Mar 2017 12:47:40 +0000