I am trying to find a simple mathematical formula to allow for a
comparison of similarity between two things composed of like and
unlike elements.  Note: This is not a homework assignment (M*A*S*H was
in primtetime the last time I wrote a university exam).

First, a simple example.  Let's say Book A has 100 chapters, and Book
B has 50 chapters.  Let's say that 20 of the chapters are common to
Books A and B.

How would you compute the "similarity factor" between Books A and B?

Second, a more relevant example.  Diagram A is made up of 100
different letter combinations (i.e. aa, ab, ac, etc.) and 100
different numbers.  Diagram B is made up of 50 different letter
combinations and 50 different numbers.  Let's say that 25 of the
letter combinations, and 10 of the numbers were common to both Diagram
A and B.

How would you compute a "similarity factor" between Diagrams A and B? 
Basically, I would like to know if anyone can come up with a
mathematical formula that yields a number that one can use to assess
how similar these diagrams are to each other.

Note: Unless someone can convince me otherwise, what I am looking for
is a formula for how similar the diagrams are to each other, not how
similar Diagram A is to Diagram B and vice versa.

If you are comparing something with agreed upon basic units it would
be simple.  You could pick out the units (squares of the diagram for
instance, pixels etc...) and compare them. The trick is choosing how
to subdivide the thing you want to compare.

You were looking at books.  You could deal with words, with bits of info etc...

One thing you could look into is compression technologies, such as
picture compression and zip archiving programs.  They tend to look for
regularities in the  things they are compressing so they can save
space by saying "just repeat that bit there' instead of detailing the
whole thing.

You might get a measure by comparing the difference between the
compression ratios of a file containing two instances of the same,
versus one containing one of each.  You would hope that the ratio
would be the same whatever two instances of the same you chose, and
that the more different the things are, the less compression you would
get.  Try looking into who engineers compression packages,and the
maths that they use to do it,  if you want to find a more formal
answer.