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


It seems that the instructor was explaining that the entire set does not have a least element. You are correct in saying that the subset above has a least element: aab. But because we can always construct an element smaller than that in the set (it's not in the subset though), the entire infinite set does not have a least element.
