Lattice

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 _____.

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) . Then

Consider the following Hasse diagrams.
i.

ii.

iii.

iv.

Which all of the above represent a lattice?

The following is the Hasse diagram of the poset [{a,b,c,d,e}, \prec ]

The poset is:

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?