What is Abstract Algebra and why is it important?

[StephenPrivitera]

I downloaded that book from the physics Napster. It looks like something I should be able to handle, but I need help understanding some of the notation.

"Any set called an index set is assumed to be non-void. Suppose T is an index set and for each t within T, A_t is a set.

[inter] A_t = {x : if t E T, x E A_t}
t E T

= for each t within T, x is an element of A_t"

There is also another expression with Union rather than Intersection, which I find more difficult to understand (U A_t with tET written below U).
U = there exists t E T with x E A_t

What is an index set?
Can someone explain this notation?
What is "t" - a number? a set? What does "there exists t E T with x E A_t" mean?

 

[Hurkyl]

It might help to see an examlpe of what is being generalized.


What is the definition of the intersection of two sets? It is:

A [inter] B = {x : x is in A and x is in B}

Changing notation slightly, this is the same as:

A_1 [inter] A_2 = {x : x is in A_1 and x is in A_2}
= {x : [uni] t in {1, 2}, x is in A_t}

(the underscore is meant to represent subscripts. I'm lazy today and don't want to write a million tags to do it properly)


To use some fancy terminology, we are taking the intersection of a collection A of sets indexed by the set {1, 2}.

IOW we take the intersection of the sets A_1 and A_2.


What if we had more sets? Well:

A [inter] B [inter] C = {x : x is in A and x is in B and x is in C}

rewriting again:

A_1 [inter] A_2 [inter] A_3 = {x : [uni] t in {1, 2, 3}, x is in A_t}

We're motivated to write this repeated intersection similar to how we write sums with Σ notation:

A_1 [inter] A_2 [inter] ... [inter] A_n = [inter]_(t=1..n) A_t

and we can write the definition:

[inter]_(t=1..n) A_t = {x : x is in A_1 and x is in A_2 and ... and x is in A_n}
= {x : [uni] t in {1, 2, ..., n}, x is in A_t}


The set {1, 2, ..., n} is called an index set because it's the set of all indices (aka subscripts) for the sets in the collection A.


So now we see how to write the general definition. If we have a collection A of sets indexed by the set T, we can define the intersection of the sets in A as:

[inter]_(t in T) A_t ={x : [uni] t in T, x is in A_t}


In general, T may be any set.



For union, the idea is the same, we just need to figure out how to generalize. Notice that:

x is in A_1 or x is in A_2 or ... or x is in A_n

means the same thing as

there exists a t in {1, 2, ..., n} such that x is in A_t


If that's not clear, maybe a small example will help:

x is in A_1 or x is in A_2

is the same as:

x is in A_t where t = 1 or t = 2

is the same thing as

x is in A_t where t is in {1, 2}

which implies

there exists a t in {1, 2} such that x is in A_t

 

[StephenPrivitera]

The set {1, 2, ..., n} is called an index set because it's the set of all indices (aka subscripts) for the sets in the collection A.
________
This is the part I missed. I didn't realize this was a generalized result. To make sure I understand:
If we say x in an element of the intersection of the sets A_1 ,... ,A_t, then we know that x is an element of every set from A_1 to A_t. And if we say that x is an element of the union of those sets, then we know x is an element of at least one of those sets. Correct me if I'm wrong. I think using these words makes the notation much easier to understand.
BTW, do you suppose there may be some prerequisites to consider before reading this book? As far as my experience goes, I've studied two semesters of calculus with some differential equations.

 

[Hurkyl]

Yes, your rephrasing is correct! Yay!

One thing I would like to emphasize is that the index set does not necessarily have a sequential order to it like the natural numbers... for instance I might want to use the real numbers as my index set. The distinction doesn't matter so much with the cases at hand, but it can later on.



Abstract algebra is a sort of foundation subject. It essentially starts from scratch... the problem is that it's, well, abstract!

If you want my opinion, you should probably look into Linear Algebra first. It's a little closer to what you already know, is almost immediately practical, and if you get a good text that also talks about abstract vector spaces (or general vector spaces), you'll get your first introduction to abstraction, but be able to see immediate applications of the abstract point of view.

Another benefit you are likely to get out of Linear Algebra is more familiarity with proofs... which are very important to be able to understand for more advanced mathematics. A whole course on proofs is probably a good idea, and is probably a prerequisite for any advanced mathematics course at your university. A math advisor... or any professor really... is probably a good person to talk too.


Of course, if you've developed the right skills and interests, an abstract algebra could possibly be the right course for you too.