Lattice
Q1.
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) . ThenQ2.
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 _____.Q3.
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.
Consider the following Hasse diagrams.i.ii.iii.iv.Which all of the above represent a lattice?Q5.
In the lattice defined by the Hasse diagram given in following figure, how many complements does the element 'e' have?Q6.
The following is the Hasse diagram of the poset [{a,b,c,d,e}, \prec ] The poset is: