Lattice
Q1.
Consider the following Hasse diagrams.i.ii.iii.iv.Which all of the above represent a lattice?Q2.
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) . ThenQ3.
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?Q4.
In the lattice defined by the Hasse diagram given in following figure, how many complements does the element 'e' have?Q5.
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 _____.Q6.
The following is the Hasse diagram of the poset [{a,b,c,d,e}, \prec ] The poset is: