Puzzle of the Day

A lattice is a mathematical structure composed of a set S and a relation r. If x, y, and z are elements of S, and r is a 
relation over those elements, then (S, r) is a lattice if and only if the following conditions are met:

1.	r is reflexive, so for every element x in S, xrx.

2.	r is antisymmetric, so if xry and yrx, then x = y.

3.	r is transitive, so if xry and yrz, then xrz.

4.	Every pair of elements in S has a lower bound. In other words, there is a z in S such that zrx and zry for every pair 
(x, y).

5.	Every pair of elements in S has an upper bound. In other words, there is a z in S such that xrz and yrz for every 
pair (x, y).

Does ({ 1, 2, 3, 4, 5 }, ?) form a lattice? How about ({true, false}, and)?