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A few mathematical study skills... Reading Definitions

by Ashley Reiter Ahlin

Nearly everyone knows (or thinks they know) how to read a novel, but
reading a math book is quite a different thing. To begin with, there are
all these definitions! And it's not always clear why one would care to
know about these things being defined. So what should you do when you
read a definition?
Ask yourself (or the book) a few questions:

- What kind of creature does the definition apply to? integers? matrices?
sets? functions? some pair of these together?
- How do we check to see if it's satisfied? (How would we prove that
something satisfied it?)
- Are there necessary or sufficient conditions for it? That is, is there
some set of objects which I already understand which is a subset or a
superset of this set?
- Does anything satisfy this definition? Is there a whole class of things
which I know satisfy this definition?
- Does anything not satisfy this definition?
- What special properties do these objects have, that would motivate us to
make this definition?
- Is there a nice classification of these things?

Let's apply this to an example, abelian groups:
- What kind of creature does it apply to? Well, to a group ... that is,
to a set together with a binary operation.
- How do we check to see if it's satisfied? The startling thing is that
we have to compare every single pair of elements! This would be a big
job, so we'd like to have some other sufficient conditions ... can you
find any?
- Are there necessary or sufficient conditions for it? Well, it's
sufficient that the group be cyclic, as we saw in the homework. Do you
know of any necessary conditions?
- Does anything satisfy this definition? Well, yes ... the group of
integers under addition for example. We have a whole class of things
which satisfy the definition, too -- cyclic groups.
- Does anything not satisfy this definition? Yes, matrix groups come to
mind first. There ARE finite non-abelian groups, but this is harder to
see ... do you know of one yet?
- What special properties do these objects have, that would motivate us to
make this definition?
Some of these properties are obvious, others are things which we
had to prove. One example is that: If H and K are subgroups of an
abelian group, then HK is also a subgroup.
- Is there a nice classification of these things? Why yes! We'll get to
it later ... it says, basically, that a finite abelian group is always
built in a simple way from cyclic groups -- Z and Z_n.

Reading
Mathematical Theorems

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