TITLE: How many integers in the range [1,999] are divisible by exactly 1 of 7 and 11?
QUESTION [6 upvotes]: This is a question in Kenneth Rosen's Discrete Mathematics textbook 6th edition.  I haven't had trouble with any other counting problems regarding "how many numbers in range [x,y] have divisibility property Z?"  My issue is I have no idea what Rosen is asking for, i.e. I don't understand the question because I don't know what he wants me to compute.  
Therefore this is not a duplicate of this question (1) https://math.stackexchange.com/questions/588160/how-many-positive-integers-less-than-1000-are-divisible, since while the answer is given, it doesn't explain the language of the question and what is being computed.  I have no idea why the number of integers divisible by 7 or 11 minus the number of integer divisible by 77 (11 and 7) is the answer to this question.  Both of these values I've already computed correctly (in separate questions).  
In context:
20.  How many positive integers less than 1000
e) are divisible by exactly one of 7 and 11?
Thus my question is: what/which numbers am I supposed to count/compute?

REPLY [4 votes]: This question can be solved using PIE or Principle of Inclusion and Exclusion, which is a very useful technique in the field of combinatorics. 
For the question you've got here, we let $A$ (see Venn Diagram below) be the set of natural numbers less than $1000$ that is divisible by $7$; whereas let  $B$ be the set of natural numbers less than $1000$ that is divisible by $11$.
From the diagram, it is clear that we want the areas of $A + B - A \cap B$, where $A \cap B$ refers the intersection of sets $A, B$. Therefore, the number of integers divisible by $7$ or $11$ minus the number of natural numbers divisible by $77$ ($11$ and $7$) is the answer to the question. 

In general, for the question asking how many natural numbers less than $k \in \mathbb{Z}$ which is divisible by the integers in the set ${a_1, a_2, \ldots a_n}$, we can represent the question in terms of $n$ sets (or circles in the Venn diagram) and then solve from there. 
P.S. How can I adjust the size of the image???