Discrete Mathematics
Q111.
In a room containing 28 people, there are 18 people who speak English, 15, people who speak Hindi and 22 people who speak Kannada. 9 persons speak both English and Hindi, 11 persons speak both Hindi and Kannada whereas 13 persons speak both Kannada and English. How many speak all three languages?Q114.
The following is the incomplete operation table of a 4-element group. The last row of the table isQ115.
Let A be the set of all non-singular matrices over real number and let * be the matrix multiplication operation. ThenQ116.
The line graph L(G) of a simple graph G is defined as follows: There is exactly one vertex v(e) in L(G) for each edge e in G. For any two edges e and e' in G, L(G) has an edge between v(e) and v(e'), if and only if e and e' are incident with the same vertex in G. Which of the following statements is/are TRUE? (P) The line graph of a cycle is a cycle. (Q) The line graph of a clique is a clique. (R) The line graph of a planar graph is planar. (S) The line graph of a tree is a tree.Q118.
Suppose L={p,q,r,s,t} is a lattice represented by the following Hasse diagram: For any x,y\in L not necessarily distinct, x\vee y and x\wedge y are join and meet of x,y respectively. Let L^{3}=\{(x,y,z):x,y,z\in L\} be the set of all ordered triplets of the elements of L. Let P_{r} be the probability that an element (x,y,z)\in L^{3} chosen equiprobably satisfies x\vee (y \wedge z)=(x\vee y)\wedge (x\vee z) . ThenQ119.
Consider the set X={a, b,c,d,e} under the partial ordering R={(a,a),(a,b),(a,c),(a,d),(a,e),(b,b),(b,c),(b,e),(c,c),(c,e),(d,d),(d,e),(e,e)}. The Hasse diagram of the partial order (X, R) is shown below. The minimum number of ordered pairs that need to be added to R to make (X, R) a lattice is _____.Q120.
The inclusion of which of the following sets into S = {{1, 2}, {1, 2, 3}, {1, 3, 5}, {1, 2, 4}, {1, 2, 3, 4, 5}} is necessary and sufficient to make S a complete lattice under the partial order defined by set containment?