GATE Overflow - Recent questions and answers in Set Theory & Algebra
https://gateoverflow.in/qa/mathematics/discrete-mathematics/set-theory-%26-algebra
Powered by Question2AnswerAnswered: GATE2007-3
https://gateoverflow.in/1202/gate2007-3?show=159969#a159969
<p>The number of m-ary functions in p-valued algebra having n-variables is given by $m^{p^{n}}$
<br>
<br>
Here, m = 2 (boolean functions), p = 2 (boolean variables) and n = n (number of variables)
<br>
<br>
So, total functions = $2^{2^{n}}$
<br>
<br>
<strong> Alternative approach :</strong>
<br>
<br>
With n boolean variables, we can have ${2^{n}}$ combinations for functions. And now since each of these is designed to be a boolean function, it's output value can be either 1 or 0, i.e. 2 choices for each function. So total such functions = $2^{2^{n}}$</p>Set Theory & Algebrahttps://gateoverflow.in/1202/gate2007-3?show=159969#a159969Sat, 14 Oct 2017 13:31:32 +0000Answered: GATE1993_17
https://gateoverflow.in/2314/gate1993_17?show=159607#a159607
explain with any shortcut method ...!Set Theory & Algebrahttps://gateoverflow.in/2314/gate1993_17?show=159607#a159607Fri, 13 Oct 2017 05:15:12 +0000Answered: Rossen: How to perform Composition on Directed Graph.
https://gateoverflow.in/159587/rossen-how-to-perform-composition-on-directed-graph?show=159602#a159602
<p>Given R and S relations , the composition R o S is defined as :</p>
<blockquote>
<p>R o S = { (x,z) | (x,y) ∈ S and (y,z) ∈ R }</p>
</blockquote>
<p> So for the two directed graphs , we can write equivalent R and S relations whose pairs are the edges in R and S respectively..Having written the R and S relations , we can do the composition R o S as mentioned above and obtain the resultant graph.</p>
<p><em><strong>This way composition of graphs is done.</strong></em></p>Set Theory & Algebrahttps://gateoverflow.in/159587/rossen-how-to-perform-composition-on-directed-graph?show=159602#a159602Fri, 13 Oct 2017 04:58:18 +0000Answered: ISRO2017-9
https://gateoverflow.in/128555/isro2017-9?show=159497#a159497
A={1,2,3,4,5,6,7,8} and B={1,3,5,6,7,8,9}<br />
<br />
A U B = {1,2,3,4,5,6,7,8,9} , A ∩ B = {1,3,5,6,7,8}<br />
<br />
Symmetric difference = (A U B) - (A ∩ B) = {2,4,9}<br />
<br />
option BSet Theory & Algebrahttps://gateoverflow.in/128555/isro2017-9?show=159497#a159497Thu, 12 Oct 2017 13:14:11 +0000Answered: GATE2014-1-50
https://gateoverflow.in/1930/gate2014-1-50?show=159468#a159468
<p>f:{0,1}<sup>4</sup>→{0,1} </p>
<p>{0,1}<sup>4 </sup>contains total 2<sup>4</sup> elements<sup> </sup></p>
<p>so 16 elements and S will be [co-domain]<sup>domain </sup>= 2<sup>16</sup></p>
<p>N is S to {0,1} so 2^2<sup>16</sup></p>
<p>log<sub>2</sub>log<sub>2</sub>N = 16</p>
<p> </p>Set Theory & Algebrahttps://gateoverflow.in/1930/gate2014-1-50?show=159468#a159468Thu, 12 Oct 2017 11:55:41 +0000Answered: GATE2015-2_GA_3
https://gateoverflow.in/8030/gate2015-2_ga_3?show=159462#a159462
There is three values of x = -1,0,1<br />
<br />
1-|x| =1-|-1| = 0<br />
<br />
1-|x| = 1- |0|=1<br />
<br />
1-|x| = 1-|1| =0<br />
<br />
so option C 0,1Set Theory & Algebrahttps://gateoverflow.in/8030/gate2015-2_ga_3?show=159462#a159462Thu, 12 Oct 2017 11:37:16 +0000Answered: GATE1993_8.6
https://gateoverflow.in/2304/gate1993_8-6?show=159459#a159459
<p>Answer is <sup>n</sup>P<sub>m</sub></p>Set Theory & Algebrahttps://gateoverflow.in/2304/gate1993_8-6?show=159459#a159459Thu, 12 Oct 2017 11:17:05 +0000Answered: GATE2015-2_54
https://gateoverflow.in/8257/gate2015-2_54?show=159457#a159457
<p>Total funtions from X to Y = [Order(Y) ]<sup>order(x)</sup></p>
<p><sup>20</sup>C<sub>1</sub> * <sup>20</sup> C<sub>2</sub>/ 20<sup>2 </sup></p>
<p>= 0.95</p>Set Theory & Algebrahttps://gateoverflow.in/8257/gate2015-2_54?show=159457#a159457Thu, 12 Oct 2017 11:12:54 +0000Answered: GATE2015-1_5
https://gateoverflow.in/8025/gate2015-1_5?show=159456#a159456
Correct answer is A because of <br />
<br />
(x/x-1)*(1/1-x) <br />
<br />
g(x) = 1-x and h(x)=x/x-1<br />
<br />
so h(x)/g(x)Set Theory & Algebrahttps://gateoverflow.in/8025/gate2015-1_5?show=159456#a159456Thu, 12 Oct 2017 11:06:36 +0000Answered: GATE2003-39
https://gateoverflow.in/930/gate2003-39?show=159309#a159309
<p>Since no sequence is given we need the help of options to identify the correct encoding.</p>
<p>
<br>
<em>h</em>(⟨<em>s</em><em>i</em>…<em>s</em><em>n</em>⟩)=Π<em>n</em><em>i</em>=1<em> P</em><em>i^f</em>(<em>s</em><em>i</em>) =P1^f(s1) * P2^f(s2) * .....*Pn^f(sn)
<br>
P1=2, P2=3, P3=5, P4=7 and so on...</p>
<p>g(ai)={3,5,7,9,11}.</p>
<p>Since in all the options there are first 3 prime nos. given, we can conclude that the sequence s goes up to n=3.</p>
<p>Now, f(<em>s</em>)=Π<em>n</em><em>i</em>=1<em>Pi^</em><em>g</em>(<em>a</em><em>i</em>)<em>= </em>P1^g(s1) * P2^g(s2) * .....*Pn^g(sn). As it starts with P1 which is 2 and g(ai)≠0 for any case, so the value of f(s) can never be odd. Thus we can eliminate option A and C.</p>
<p>Next we see the powers of 2 in option B and D. Concentrating only on the power of 2 we find---></p>
<p>B shows 8 which can be obtained if for i=1, g(a1)=3 and P1=2 we already know. 2^3=8.
<br>
C shows 10 which can be obtained if for i=1, g(a1)=1 and P1=2 (we know) , for i=2,g(a2)=0 and P2=3 (we know), for i=3, g(a3)=1 and P3=5 (we know). Then (2^1)*(3^0)*(5^1)=10. But we know that g(ai)≠0 for any case. So C can't be the answer.</p>
<p>Hence B.</p>
<p> </p>Set Theory & Algebrahttps://gateoverflow.in/930/gate2003-39?show=159309#a159309Wed, 11 Oct 2017 19:09:00 +0000Answered: GATE2005-44
https://gateoverflow.in/1170/gate2005-44?show=159277#a159277
<p><strong>(a≡c mod 3)</strong> means remainder values 'a' can get when 'c' is divided by 3. It is {0,1,2}.</p>
<p><strong>(b≡d mod 5)</strong> remainder values 'b' can get when 'd' is divided by 5. It is {0,1,2,3,4}.</p>
<p>Whatever be the values of c and d, we can at most get 15 combinations of a and b.( i,e (0,0) (0,1) (0,2) (0,3) (0,4) ......(2,0) (2,1) (2,2) (2,3) (2,4) total 5*3=15.</p>
<p>As here we need to find the <strong>minimum</strong> number of ordered pairs, so taking any value for (c,d) will do. Therefore we consider 1 for (c,d).</p>
<p>Total number of ordered pairs become 15+1=16.</p>Set Theory & Algebrahttps://gateoverflow.in/1170/gate2005-44?show=159277#a159277Wed, 11 Oct 2017 16:02:55 +0000Answered: he number of students who take both the subjects mathematics and chemistry is 30
https://gateoverflow.in/158530/number-students-take-subjects%C2%A0mathematics%C2%A0and-chemistry?show=158847#a158847
| M and C | = 30<br />
<br />
|M| = 30*100/10 = 300<br />
<br />
|C| = 30*100/12 = 250<br />
<br />
students take at least one of these = students taking M + students taking C - students taking both<br />
<br />
= 300+250-30 = 520Set Theory & Algebrahttps://gateoverflow.in/158530/number-students-take-subjects%C2%A0mathematics%C2%A0and-chemistry?show=158847#a158847Tue, 10 Oct 2017 03:49:14 +0000Answered: GATE1998_1.8
https://gateoverflow.in/1645/gate1998_1-8?show=158505#a158505
m--->n<br />
<br />
ans: CSet Theory & Algebrahttps://gateoverflow.in/1645/gate1998_1-8?show=158505#a158505Mon, 09 Oct 2017 05:15:14 +0000Answered: GATE1998_1.7
https://gateoverflow.in/1644/gate1998_1-7?show=158503#a158503
Ans: CSet Theory & Algebrahttps://gateoverflow.in/1644/gate1998_1-7?show=158503#a158503Mon, 09 Oct 2017 05:13:55 +0000Answered: GATE1988-2xviii
https://gateoverflow.in/94353/gate1988-2xviii?show=158065#a158065
<p>See if this could help!
<br>
<img alt="" height="1003" src="http://gateoverflow.in/?qa=blob&qa_blobid=4022443249242960857" width="565"></p>Set Theory & Algebrahttps://gateoverflow.in/94353/gate1988-2xviii?show=158065#a158065Sat, 07 Oct 2017 10:23:43 +0000Answered: Inverse of a function
https://gateoverflow.in/158002/inverse-of-a-function?show=158022#a158022
<p>For a function f to be bijective, we need to check 2 conditions ~</p>
<ol>
<li>It must be one-one</li>
<li>It must be onto</li>
</ol>
<p><em>Also, a function is invertible iff it is one-one. </em></p>
<p><em>So yes, a bijective function is invertible and bijective as well</em></p>
<p>Consider this screenshot from Rosen</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=9537627575657298451"></p>
<p>You can also try an example and confirm.</p>Set Theory & Algebrahttps://gateoverflow.in/158002/inverse-of-a-function?show=158022#a158022Sat, 07 Oct 2017 06:37:30 +0000Answered: GATE1994_3.9
https://gateoverflow.in/2495/gate1994_3-9?show=157992#a157992
It is TRUE.Set Theory & Algebrahttps://gateoverflow.in/2495/gate1994_3-9?show=157992#a157992Sat, 07 Oct 2017 05:18:17 +0000Answered: GATE2015-3_23
https://gateoverflow.in/8426/gate2015-3_23?show=157554#a157554
(1) X \ Y = X - Y<br />
<br />
(2) Y' \ X' = Y' - X' = (U - Y) - (U - X) = X - Y (U represents universal set for X and Y. Here it is power set.)<br />
<br />
Equation (1) is equal to (2) So answer is (D) Part.Set Theory & Algebrahttps://gateoverflow.in/8426/gate2015-3_23?show=157554#a157554Thu, 05 Oct 2017 13:51:19 +0000Answered: Relations
https://gateoverflow.in/157250/relations?show=157409#a157409
Number of symmetric relations which are not reflexive<br />
<br />
{ }<br />
<br />
{(a,a)}<br />
<br />
{(b,b)}<br />
<br />
{(a,b),(b,a)}<br />
<br />
{(a,a),(a,b),(b,a)}<br />
<br />
{(a,b),(b,a),(b,b)}<br />
<br />
Hence I think 6 should be the correct ans.Set Theory & Algebrahttps://gateoverflow.in/157250/relations?show=157409#a157409Thu, 05 Oct 2017 05:57:06 +0000Answered: Discrete-mathematics
https://gateoverflow.in/157333/discrete-mathematics?show=157360#a157360
<p><strong>option 1</strong> - suppose A has elements 1,2,3 .......now A is mapped one to one to A itself</p>
<p>lets consider mapping 1-1 , 2-2 , 3-3 ...every element has atleast one mapping which means its onto</p>
<p><strong>option 2</strong> - is the same</p>
<p><strong>option 3</strong> - when a set is given....the power set is taken from the given set with various combinations which we know...so if its countably infinite then power set will also be countably infinite</p>
<p><strong>ANS - All are true</strong></p>Set Theory & Algebrahttps://gateoverflow.in/157333/discrete-mathematics?show=157360#a157360Thu, 05 Oct 2017 04:02:02 +0000Answered: Set Theory and algebra
https://gateoverflow.in/157194/set-theory-and-algebra?show=157299#a157299
<ol>
<li>For T ⊆ S it will be <sup>25</sup>C<sub>5 </sub></li>
<li>For T ⊆ P(S ) it will be <sup>2^25</sup>C<sub>5 </sub></li>
</ol>
<p>Correct me if I am wrong!!</p>Set Theory & Algebrahttps://gateoverflow.in/157194/set-theory-and-algebra?show=157299#a157299Wed, 04 Oct 2017 18:55:40 +0000Answered: Set theory
https://gateoverflow.in/157200/set-theory?show=157261#a157261
<p>Here X and Y are representing subsets such that X ∩ Y = Φ meaning that X and Y are disjoint subsets..So for finding number of such pairs which are disjoint to each other and let the size of the set = n.</p>
<p>Let we have number of elements in X = r..</p>
<p>So number of ways of selecting elements in X such that it contains 'r' elements = <sup>n</sup>C<sub>r</sub></p>
<p>So we have number of elements left for inclusion in Y = n-r..</p>
<p>Now each of these 'n-r' elements can be either selected or rejected , hence 2 choices for each element..</p>
<p>Hence total number of possibilities of Y such that it is disjoint to X = 2 . 2 . ......(n-r) times</p>
<p> = 2<sup>n-r</sup></p>
<p>Hence number of pairs (X,Y) such that X contains 'r' elements = <sup>n</sup>C<sub>r </sub>* 2<sup>n-r</sup></p>
<p>Now 'r' varies from 0 to n..</p>
<p>Hence total number of pairs (X,Y) = Σ <sup>n</sup>C<sub>r </sub>* 2<sup>n-r</sup></p>
<p> = (1 + 2)<sup>n</sup></p>
<p> = 3<sup>n</sup></p>
<p>But (X,Y) is same as (Y,X) , hence double counting is to be avoided..</p>
<p>Hence number of such pairs = [ ( 3<sup>n </sup>- 1 ) / 2 ] + 1</p>
<p><em><strong>Substituting n = 6 , we get number of pairs = 728 / 2 + 1</strong></em></p>
<p><em><strong> = 365</strong></em></p>Set Theory & Algebrahttps://gateoverflow.in/157200/set-theory?show=157261#a157261Wed, 04 Oct 2017 16:52:56 +0000Answered: GATE2005-IT-31
https://gateoverflow.in/3777/gate2005-it-31?show=157124#a157124
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=6793386542531110012"></p>
<p>If condition p --> q is true then its contrapositive ( ~q --> ~p ) is also true</p>
<p>above example is if g is not on to then 'h' is also not onto</p>
<p>so its contrapositive is also true that is { h is onto --> g is onto}</p>
<p> </p>Set Theory & Algebrahttps://gateoverflow.in/3777/gate2005-it-31?show=157124#a157124Wed, 04 Oct 2017 06:13:08 +0000set theory and algebra
https://gateoverflow.in/157037/set-theory-and-algebra
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=7506087194220041859"></p>Set Theory & Algebrahttps://gateoverflow.in/157037/set-theory-and-algebraTue, 03 Oct 2017 17:09:38 +0000Answered: relation doubt
https://gateoverflow.in/157010/relation-doubt?show=157014#a157014
<p>We know :</p>
<blockquote>
<p>Relation R of a set X to Y in general is a subset of the set of ordered pairs resulted by cartesian product of sets A and B denoted by A * B..But a relation has to satisfy a given condition..</p>
</blockquote>
<p>So here relation is defined on A --> A such that only such ordered pairs will be included which have x = y.</p>
<p>Hence possible ordered pairs = (0,0) , (1,1) , (2,2) , (3,3) ..</p>
<p>Now each of these can either be included or rejected hence we have 2 choices for each..</p>
<p>Hence number of relations = 2 * 2 * 2 * 2</p>
<p> <em><strong>= 16 [ Including null relation as in null relation , no pair hence no issue of equality , so can be considered ] </strong></em></p>Set Theory & Algebrahttps://gateoverflow.in/157010/relation-doubt?show=157014#a157014Tue, 03 Oct 2017 16:20:54 +0000Answered: #GroupTheory #TestBook
https://gateoverflow.in/156768/%23grouptheory-%23testbook?show=156776#a156776
<p>Here we are given that the order of the group is 10 and a is the generator..</p>
<p>Hence a<sup>10</sup> = e where e is the identity element but we dont know about a<sup>8</sup>..Hence we have to write it in terms of a<sup>10</sup> as that is a known result.</p>
<p>Let we have y = a<sup>40</sup> which can be written as : a<sup>40</sup> = (a<sup>10</sup>)<sup>4</sup> = e<sup>4</sup> = e ...(1) </p>
<p> Also , a<sup>40</sup> = (a<sup>8</sup>)<sup>5</sup> = e [ Follows from (1) ]</p>
<p>Hence it means that a<sup>8</sup> is repeated 5 times in order to get the identity element e which is least number of times to do so.</p>
<p>Also from the corollary of the Lagranges' Theorem , </p>
<blockquote>
<p>Order of an element divides the order of the order of the group (The actual theorem is order of a subgroup divides the order of the group)</p>
</blockquote>
<p>Here 5 divides the order of group i,e, 10..</p>
<p><em><strong>Hence order of a<sup>8</sup> = 5 </strong></em></p>Set Theory & Algebrahttps://gateoverflow.in/156768/%23grouptheory-%23testbook?show=156776#a156776Mon, 02 Oct 2017 15:50:50 +0000Answered: #GroupTheory #TestBook
https://gateoverflow.in/156767/%23grouptheory-%23testbook?show=156769#a156769
<p>For existence identity element 'e' : </p>
<p>x op e = e op x = x</p>
<p>Here x op y = x + y - 2</p>
<p> So x op e = x + e - 2 = x [ As mentioned above ]</p>
<p> ==> e = 2</p>
<p>Now e op x should also yield the same identity element , else identity element does not exist..</p>
<p>So e op x = e + x - 2 = x</p>
<p> ==> e = 2</p>
<p>Hence in both cases we get e = 2 and hence 2 is the required identity element..</p>
<p>Now for finding inverse , say y w.r.t x :</p>
<p> x op y = e</p>
<p>==> x + y - 2 = 2</p>
<p>==> y = 4 - x</p>
<p> </p>
<p><strong><em>Hence C) should be the correct answer.. </em></strong></p>
<p><strong><em> </em></strong></p>Set Theory & Algebrahttps://gateoverflow.in/156767/%23grouptheory-%23testbook?show=156769#a156769Mon, 02 Oct 2017 15:22:49 +0000Answered: GATE2007-IT-80
https://gateoverflow.in/3532/gate2007-it-80?show=156755#a156755
<p>Answer is (A) part.</p>
<p>Please refer --><a rel="nofollow" href="https://stackoverflow.com/questions/8222108/max-slope-from-set-of-points">https://stackoverflow.com/questions/8222108/max-slope-from-set-of-point</a></p>Set Theory & Algebrahttps://gateoverflow.in/3532/gate2007-it-80?show=156755#a156755Mon, 02 Oct 2017 13:53:58 +0000Group Theory
https://gateoverflow.in/156722/group-theory
<p>Let (G,<em>star) be a group such that for any a,b belongs to G, (a</em>b)power^71==(a)power^71 * (b)power^71, (ab)power^72==(a)power^72 * (b)power^72 and (ab)power^73=(a)power^73 * (b)power^73. Proove that G is Abelian Group.</p>Set Theory & Algebrahttps://gateoverflow.in/156722/group-theoryMon, 02 Oct 2017 10:56:28 +0000Answered: GATE1997_3.1
https://gateoverflow.in/2232/gate1997_3-1?show=156562#a156562
<p>A- For Monoid, semigroup should have identity property</p>
<p>B-For Abelian group, group should have commutative property</p>
<p>C-For Group Monoid should have a inverse property</p>
<p>But there is no property n*m=max(n,m) so </p>
<p><strong>Option D must be True</strong></p>Set Theory & Algebrahttps://gateoverflow.in/2232/gate1997_3-1?show=156562#a156562Sun, 01 Oct 2017 14:55:06 +0000Answered: UGCNET-june2006-5
https://gateoverflow.in/156107/ugcnet-june2006-5?show=156449#a156449
y=x/2 // monotonic increasing =>one-one certainly....<br />
<br />
linear polynomials are always one-one<br />
<br />
<br />
<br />
domain and range ∈ (-1,1) but here co domain ∈ (-1/2 ,1/2) so some elements won't be connected to any x..<br />
<br />
so it's not an onto function...Set Theory & Algebrahttps://gateoverflow.in/156107/ugcnet-june2006-5?show=156449#a156449Sun, 01 Oct 2017 04:14:47 +0000Answered: ace test
https://gateoverflow.in/156332/ace-test?show=156339#a156339
<p><strong>Equivalence relation:</strong> reflexive, symmetric, and transitive
<br>
<strong>Partial order relation:</strong> reflexive, antisymmetric, and transitive</p>
<p>|A| = n
<br>
|AxA| = $n^2$</p>
<p>we have total $n^2$ ordered pairs, for n diagonal pairs we don't have any choice, we have to keep them to satisfy reflexive property.
<br>
Now (n^2-n) ordered pairs are only left.
<br>
if (a,b) is present in our relation, then Equivalence relation says (b,a) should also be present as it needs to satisfy <strong>symmetric </strong>property, but partial order relation oppose it because it has to satisfy <strong>antisymmetric</strong> property.</p>
<p><strong>Hence, there will only be one such relation </strong>which is <ins>both</ins> Equivalence and partial order.</p>Set Theory & Algebrahttps://gateoverflow.in/156332/ace-test?show=156339#a156339Sat, 30 Sep 2017 11:34:38 +0000Answered: TIFR-2011-Maths-A-1
https://gateoverflow.in/30174/tifr-2011-maths-a-1?show=155858#a155858
OscillatesSet Theory & Algebrahttps://gateoverflow.in/30174/tifr-2011-maths-a-1?show=155858#a155858Thu, 28 Sep 2017 12:47:41 +0000Answered: UGCNET-dec2008-ii-12
https://gateoverflow.in/155035/ugcnet-dec2008-ii-12?show=155424#a155424
Ans: DSet Theory & Algebrahttps://gateoverflow.in/155035/ugcnet-dec2008-ii-12?show=155424#a155424Tue, 26 Sep 2017 15:32:33 +0000Answered: Set |Subset | Volume 1 Q6
https://gateoverflow.in/51833/set-subset-volume-1-q6?show=154651#a154651
But in question why they mention total no of subset of x such that a exor b={2,3,....100}<br />
<br />
Why not they start from 1Set Theory & Algebrahttps://gateoverflow.in/51833/set-subset-volume-1-q6?show=154651#a154651Sun, 24 Sep 2017 03:37:32 +0000Answered: Function
https://gateoverflow.in/152643/function?show=154285#a154285
<p>4<sup>5</sup>-(<sup>4</sup>c<sub>1</sub>)3<sup>5</sup>+(<sup>4</sup>c<sup><sub>2</sub></sup>)2<sup>5</sup>-(<sup>4</sup>c<sub>3</sub>)1<sup>5</sup></p>
<p>=>1024-972+192-4
<br>
=>240</p>Set Theory & Algebrahttps://gateoverflow.in/152643/function?show=154285#a154285Fri, 22 Sep 2017 16:09:04 +0000graph theory
https://gateoverflow.in/153771/graph-theory
<p><strong>"A planar graph need not to be connected" </strong></p>
<p>Can someone plz explain with an example .</p>Set Theory & Algebrahttps://gateoverflow.in/153771/graph-theoryWed, 20 Sep 2017 18:33:56 +0000Answered: GATE1996_1.3
https://gateoverflow.in/2707/gate1996_1-3?show=153672#a153672
<p>How to know whether 97 is a prime or not ? </p>
<p>Take sqrt of 97 = 10 (approx)</p>
<p>now check whether 97 is divisible by all prime numbers which are <= 10 </p>
<p>If no then it is a prime number ... 97 is not divisible by 2,3,5,7 .. so 97 is a prime number...</p>
<p>So 97 cannot be written as a product of 2 numbers(apart from 97*1)...And so 97 cannot be represented as n<sup>m </sup>, (apart from 97<sup>1</sup> ) ...</p>
<p>So only way we could have got 97 functions is using 97<sup>1</sup> which is possible only if 1 element is in domain and 97 elements in co-domain.</p>
<p>B) says number of fuinctions is 1.</p>
<p>C) says number of functions is 97<sup>97</sup>. </p>Set Theory & Algebrahttps://gateoverflow.in/2707/gate1996_1-3?show=153672#a153672Wed, 20 Sep 2017 11:17:09 +0000Answered: TIFR2013-B-16
https://gateoverflow.in/25859/tifr2013-b-16?show=153659#a153659
<p><strong>T<sub>k,n </sub>= 1</strong> iff there are <strong>k 1's</strong> out of <strong>n</strong> bit binary string.</p>
<p>Similiarly <strong>T<sub>k,n-1 </sub>= 1</strong> iff <strong>k 1's</strong> out of <strong>(n-1) bits</strong></p>
<p>It is given that in <strong><y<sub>1</sub>,y<sub>2</sub>,y<sub>3</sub>,y<sub>4</sub>,y<sub>5</sub> ....y<sub>n</sub>></strong> has exactly k 1's which means out of n bits k bits are 1's.</p>
<p>Now suddenly one bit y<sub>i</sub> is removed (that one bit can be 1 or 0, we don't know). Removing one bit will make the number of bits as (n-1).</p>
<p>So <strong>T<sub>k,n-1</sub></strong> really depends on whether i th bit which we removed is a 0 or 1. If 0, then we have k 1's in our remaining (n-1) bits and so we should return 1. If 1, then we have only (k-1) 1's out of (n-1) bits and so we need to return 0.</p>
<p>So <strong>T<sub>k,n-1</sub></strong> is <strong>NOT (y<sub>i</sub>)</strong></p>Set Theory & Algebrahttps://gateoverflow.in/25859/tifr2013-b-16?show=153659#a153659Wed, 20 Sep 2017 09:26:35 +0000Answered: Self doubt - Power Set
https://gateoverflow.in/153076/self-doubt-power-set?show=153522#a153522
<p><strong>P(A)=P(B) iff A=B</strong></p>
<p>first understand what does it mean by equal set.<strong> </strong></p>
<p>A={1,2,3} B={2,1,3} // They are equal sets, equal sets are nothing but same set is written twice(we know order of elements in set doesn't matter)</p>
<p>we can easily prove our claim</p>
<p>consider set are not equal but power set are</p>
<p>Let suppose A≠B means we have one element X that ∈ A but X∉ B so in that way {X} ∈ P(A) whereas {X} ∉ P(B)</p>
<p>But as we considered power set are equal so by contradiction we can say iff power set are equal set are equal.</p>Set Theory & Algebrahttps://gateoverflow.in/153076/self-doubt-power-set?show=153522#a153522Tue, 19 Sep 2017 16:58:24 +0000Answered: TIFR2017-A-10
https://gateoverflow.in/95272/tifr2017-a-10?show=153506#a153506
<p>If the number of elements in Co-domain is greater than number of elements of domain, then <strong>the function cannot be onto</strong>...</p>
<p>If the number of elements in domain is greater than codomain,then <strong>the function cannot be one-to-one</strong>.</p>
<p>Here number of elements in co-domain is greater than domain,so obviously there will be atleast one element in co-domain which will not have a preimage in the domain. But it can be one-to-one... </p>
<p>So <strong>Option B)</strong></p>
<p> </p>Set Theory & Algebrahttps://gateoverflow.in/95272/tifr2017-a-10?show=153506#a153506Tue, 19 Sep 2017 15:28:47 +0000Answered: Problem in sets
https://gateoverflow.in/153321/problem-in-sets?show=153335#a153335
<p><strong>S1</strong> : All the elements in A are mapped to B , therefore f is a valid function<strong> (S1 is true)</strong></p>
<p><strong>S2</strong> : All the elements in B are not mapped to C, therefore g is not a valid function<strong> (S2 is false)</strong></p>
<p><strong>S3</strong> : f(g(1)) cant be found because g(1) has no mapping therefore f 0 g is not valid<strong> ( S3 is false)</strong></p>
<p><strong>S4</strong>: g(f(1)) = g(2) = 0 <strong>(since f(1) =2 g(f(1)) becomes g(2) )</strong></p>
<p> g(f(2)) = g(3) = 1</p>
<p> g(f(3))=g(4) =1</p>
<p>A mapping is obtained for all elements , therefore g o f is valid <strong>(S4 is true)</strong></p>Set Theory & Algebrahttps://gateoverflow.in/153321/problem-in-sets?show=153335#a153335Tue, 19 Sep 2017 06:01:50 +0000Maths: functions: is given function bijective?
https://gateoverflow.in/153316/maths-functions-is-given-function-bijective
<p><strong>Question</strong> Determine whether the following function from R to R is a bijection
<br>
f(x) = $\frac{(x+1)}{(x+2)}$</p>
<p><strong>Solution:</strong>
<br>
First of all, this is not a function because f(-2) is not defined.</p>
<p>So, if the domain of the function is R - {-2} then this function is one-to-one.</p>
<p>And, if co-domain of this function is R - {1} then this function is onto.</p>
<p>So, the given function is bijection but with two above mentioned conditions. can someone please confirm?</p>
<p> </p>Set Theory & Algebrahttps://gateoverflow.in/153316/maths-functions-is-given-function-bijectiveTue, 19 Sep 2017 03:42:18 +0000Answered: Function Composition
https://gateoverflow.in/153066/function-composition?show=153245#a153245
<p>listen these all are <strong>one way true</strong>
<br>
<br>
<em><strong>1) if gof is one-one then f has to one-one
<br>
<br>
2) if gof is onto then g has to onto
<br>
<br>
3) if f is one-one and g is one-one then gof is one-one
<br>
<br>
4) if f is onto and g is onto then gof is onto</strong></em></p>Set Theory & Algebrahttps://gateoverflow.in/153066/function-composition?show=153245#a153245Mon, 18 Sep 2017 15:49:15 +0000Answered: GATE2014-2-50
https://gateoverflow.in/2016/gate2014-2-50?show=153196#a153196
<p>Statement-1) is <strong>TRUE</strong> because <strong>{ } > another subset</strong> because</p>
<p> Let <strong>S1</strong> denote any other subset of S other than { } </p>
<p> <strong>S1 - { } = S1</strong> ... Now smallest element in set difference is nothing but smallest element in S1 .. So <strong>S1 < { }</strong> whatever subset you take for S1 apart from { }...</p>
<p>Statement-2) is <strong>TRUE</strong> because <strong>S > anyother subset of S other than S</strong></p>
<p> Let <strong>S2</strong> be any other subset of S other than S</p>
<p> <strong>S - S2 = some subset S3 of S</strong> ... Now Smallest element is nothing but element in S but not in S2 .. Since smallest element is from S , <strong>S < S2 </strong>whatever values you substitute for S2..</p>
<p>So Option A) is TRUE...</p>Set Theory & Algebrahttps://gateoverflow.in/2016/gate2014-2-50?show=153196#a153196Mon, 18 Sep 2017 12:55:22 +0000Answered: GATE2015-1_16
https://gateoverflow.in/8238/gate2015-1_1-6?show=153153#a153153
<p>We use "subset" symbol to compare 2 sets ...Ex: Set A is a subset of set B iff every elements of set A is in set B.</p>
<p>We use "belongs to" symbol to compare a set and an element. Ex: Whether an element is present inside a set or not.</p>
<p>{5,{6}} is not a subset of 2<sup>A</sup> .. Because 5 is not present in 2<sup>A</sup>. {5} is actually present in 2<sup>A</sup>. </p>
<p>{5} (a set containing an element 5) is different from 5 (an element)...</p>Set Theory & Algebrahttps://gateoverflow.in/8238/gate2015-1_1-6?show=153153#a153153Mon, 18 Sep 2017 10:31:28 +0000UGCNET-Dec2009-ii-01
https://gateoverflow.in/152685/ugcnet-dec2009-ii-01
If she is my friend and you are her friend, then we are friends. Given this, the friend relationship in this context is ____________.<br />
<br />
(i) commutative (ii) transitive (iii) implicative (iv) equivalence<br />
<br />
(A) (i) and (ii)<br />
<br />
(B) (iii)<br />
<br />
(C) (i), (ii), (iii) and (iv)<br />
<br />
(D) None of theseSet Theory & Algebrahttps://gateoverflow.in/152685/ugcnet-dec2009-ii-01Sat, 16 Sep 2017 14:19:29 +0000Answered: GATE1994_2.9
https://gateoverflow.in/2476/gate1994_2-9?show=152566#a152566
<p>We cant draw a lattice with 1 element </p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=12192530825010464304"></p>
<p> </p>Set Theory & Algebrahttps://gateoverflow.in/2476/gate1994_2-9?show=152566#a152566Sat, 16 Sep 2017 07:28:37 +0000Answered: GATE2005-43
https://gateoverflow.in/1168/gate2005-43?show=152503#a152503
<p>Example to prove that g neednot be onto ...</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=2606418325987186369"></p>
<p> </p>
<p>f should be onto because f is not onto then it means there is atleast one element in C which will not have a pre-image in B. If that element in C doesnot have preimage in B, then it is impossible for it to have a preimage in A. If this happens then h will also become not onto. But in question it is given that h has to be onto.</p>Set Theory & Algebrahttps://gateoverflow.in/1168/gate2005-43?show=152503#a152503Sat, 16 Sep 2017 04:29:15 +0000Answered: TIFR2012-B-1
https://gateoverflow.in/25046/tifr2012-b-1?show=152195#a152195
is answer n ! hereSet Theory & Algebrahttps://gateoverflow.in/25046/tifr2012-b-1?show=152195#a152195Thu, 14 Sep 2017 16:26:57 +0000