unclear why subset of a lexicographic ordered set of strings is said NOT to have a least element. I am watching an excellent series on Discrete Mathematical Structures 
from IIT
At the 47:33 mark of the video the instructor constructs a set of 
strings as
S = {a^nb | n >= 0 }

That is.. any number of a's followed by a 'b'.
The instructor gives a subset of all the possible generated strings:
aab
ab
b

Then the instructor says there is no least element in the above set. 
But it seems clear to me 
(given the definitions that I reproduced in 'Background', below) 
that 'aab' is the lexicographically least element.
It comes before 'ab' and 'b'. 
Am I missing something here ?   
Note - I understand why the set as a whole is not well ordered. There are 
subsets like the set of all strings of S longer than 10 
characters which have no 
least element (since a's can repeat forever).    However for the subset given 
above, I think a least element is clearly there.  (But I've been 
known to be wrong in the past.. so I suspect I've misunderstood).
Background
Earlier in the video, the following definitions are stated:

Let $\langle A, \le \rangle$ be a poset and $B$ be a subset of $A$. Then an element $b$ of
  $B$ is  a least element of $B$ if for every element $b'$ (where $b'$ is an
  element of $B$) : $b \le b'$.
A binary relation $R$ on $A$ is well ordered if $R$ is a linear order and
  every  non-empty subset of $A$ has a least element.

Thanks in advance for your help 


*

*chris


This set must refer to $S$, not to the subset of 3 that he just gave. Any finite linearly ordered set has a minimum, as the instructor probably well knows.
So the set of strings over the alphabet $\{a,b\}$ is not a well-order, because there is a subset of those strings, namely $S$, that has no least element.
He just gave a few examples of strings that are in $S$, to give more of an idea of it, I think. 
