Discussion
Consider the following statements :
(I) Every bounded lattice is complete.
(II) Every Boolean lattice is a distributive lattice.
(III) Every complete lattice is bounded.
Then which of the following is true ?
Answer: B
Only (II) and (III) are true.
(II) holds by definition: a Boolean lattice is a complemented distributive lattice. (III) holds because a complete lattice has a join and meet of every subset, in particular of the whole lattice, which are its greatest and least elements.
(I) is false: the rationals in [0, 1] with the usual order form a bounded lattice (least element 0, greatest 1), but the set of rationals below \(1/\sqrt{2}\) has no least upper bound in it, so it is not complete. That rules out 'all are true' and the options leaving out (II) or (III).
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