Consider the set N* of finite sequences of natural numbers with a denoting that sequence a is a prefix of sequence b. Then, which of the following is true?
A. Every non-empty subset of has a greatest lower bound
B. It is uncountable
C. Every non-empty finite subset of has a least upper bound
D. Every non-empty subset of has a least upper bound
Answer:A
Consider the set N* of finite sequences of natural numbers with a denoting that sequence a is a prefix of sequence b. Then, which of the following is true?
A. Every non-empty subset of has a greatest lower bound B. It is uncountable C. Every non-empty finite subset of has a least upper bound D. Every non-empty subset of has a least upper bound
Explanation: Consider any sequence like “45, 8, 7, 2” – it can have many (infinite) least upper bounds like “45, 8, 7, 2, 5”, “45, 8, 7, 2, 1” and so on but it can have only 1 greatest lower bound – “45, 8, 7” because we are using the prefix relation. So, every non-empty subset has a greatest lower bound.