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

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.

• Hint: use induction to show that it is an increasing sequence. so the first term will be GLB. LUB is 2. – Krish Jan 12 '15 at 16:46
• You are understanding the problem just fine, but you seem to be missing the concept of partially ordered set, which, once understood, should dissipate the confusion regarding 'the relation $\leq$'. So, do you know what a partially ordered set is? – Git Gud Jan 12 '15 at 16:46

## 2 Answers

$S$ 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.

• Yes, you are asked for the least element of the set in the usual order. – Ross Millikan Jan 12 '15 at 17:09
• I don't know if you think $0 \in \Bbb N$ but let us assume so. Then $S=\{\frac 11,\frac 32,\frac 53,\frac74,\dots\}$ I can define a relation $\preceq$ by (assuming the fractions are in lowest terms) $\frac ab \preceq \frac cd$ if $b$ is even and $d$ is odd or $\leq$ if $b,d$ have the same parity. The order then is $\frac 32 \preceq \frac 74 \preceq \frac{11}6 \preceq \dots \preceq \frac 11 \preceq \frac 53 \preceq \frac 95 \preceq \dots$ This would give different answers. – Ross Millikan Jan 12 '15 at 18:12
• This is similar to the lexicographic order. You can define many orders on any set. The hard part is finding one that is easy to describe. The one I give here is total, like $\le$, in that any two elements can be compared. Another on $\Bbb N \setminus \{0\}$ is $a \preceq b$ if $a$ divides $b$. This one is not total. – Ross Millikan Jan 12 '15 at 18:40

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

• But the statement said $n \in \mathbb{N}$ – DeepSea Jan 12 '15 at 17:05