Hi, this is one of the questions from my Discrete Mathematics exam that I got wrong. I believe I answered 2 since I did not see the "not" in the question.
Which of the following statements is true?
1: The greatest lower bound of S = {c,d} lies outside S.
- False. e and f are not comparable so neither can be the greatest lower bound element of S.
2: e and b are not comparable.
- False. e and b are comparable through c. Since e < c and c < b then e < b.
3: the set {c,d} has maximal element b and minimal element f
- True. b is greater than c and d and there is no comparable element greater than b. f is lower than c and d and there is no comparable element lower than f.
4: the least upper bound of {c,d} are the elements a and b.
- False. a and b are not comparable so neither can be the least upper bound.
In question 1 my answer is False but I am unsure. Does lower bound elements of a subset of a poset have to be related to every element of the subset? if so that would mean f is the only lower bound of {c,d} and therefore also the greatest lower bound element and outside of S so 1 would be true?
In question 3 my answer is True but again I am not sure. When asked this way does that mean only from the elements inside of S in which case False because c and d are not comparable?