Largest, smallest, LUB and GLB of the relation $\leq$on $ \left\{\frac{2n + 1}{n + 1} \mid n \in \mathbb{N}\right\}$

Question

I have to fin the largest, smallest, lower/least upper bound (LUB) and greatest lower bound (GLB) element of the relation $\leq$ on the following set:

$$S = \left\{\frac{2n + 1}{n + 1} \mid n \in \mathbb{N}\right\}$$

I know more or less what is the smallest and the largest element (if any), that is the 2 elements that are part of set that are respectively the smallest and the largest of all the set.

The LUB (supremum) can be part of the set, but not necessarily, the same for the GLB (infimum).

What I am not understanding is the part when they ask for this elements of the relation $\leq$ on the set S. What exactly that means?

I am quite stupid at maths (and in general) and I am not exactly understanding the question. I could find the elements they are asking from a certain set, but what does the relation has to do with this? I know I am probably not thinking to some key point, I am sorry.

Answer 1

$S$ is a set of rational numbers. The relation $\le$ is the usual relation on the rationals. When you ask about largest and LUB, you need to specify what relation you are using to compare the numbers. We could choose a different relation on the set $S$ and the answers would change.

Answer 2

$GLB = MIN = 3/2, LUB = 2$, there is no $MAX$.