Logic, Groups, and Identities

Date: 02/25/99 at 14:55:25
From: Anna
Subject: Logic, Groups and Identities

I need to know about propositions and to be able to predict logic. I 
understand truth tables and expressions but I keep getting the proofs 
wrong. Is it possible for more than one answer to exist when proving 
things? In addition, I do not understand groups; the books I have read 
only gives me information on what conditions a group should have and 
then some questions, but I don't understand what these conditions 
actually mean. 

Thank you.

Date: 02/25/99 at 18:34:29
From: Doctor Schwa
Subject: Re: Logic, Groups and Identities

There are always infinitely many possible proofs of any true statement. 
The question is which one is 'best', and that is a subjective judgment, 
not a mathematical one.

If you write back with an example or two of proofs you have done, I can 
tell you whether they're correct or not, and if you also send in the 
ones from the solution sheet, I can tell you some of the criteria I 
might use to decide whether yours or the solution sheet's are 'better' 
proofs.

Groups are very general, so any one example will necessarily leave
things out, but here are a few examples to give you an idea:

Adding hours on the 12-hour clock is a group, because:

  adding 12 hours does nothing (there's an identity)
  adding x + (12-x) hours gets you to 12 (there's always an inverse)
  adding (3+4)+9 = 3+(4+9) = 4 either way (it's associative)

Ways to rotate and/or flip a square is a group, because:

  leaving the square alone is possible (there's an identity) 
  for every clockwise rotation there's a counterclockwise one, and
  for every flip there's doing the same flip again to reverse it
    (there are inverses), and
  it only matters which order you do them in, not where you put the
    parentheses, so it's associative.

That is, a group is a collection of "things to do" such that:

  Doing nothing is possible.
  Everything you do can be undone.
  The order you do things in might matter, but how you group them
    together doesn't matter.

One last property is:

Any combination of things you can do is another (single) thing you can 
do. That is, adding 4 hours and then 7 hours is the same as adding 11 
hours. Rotating the square 90 degrees and then 180 degrees is the same 
as rotating it 270 degrees.

Does that help?

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/   

Date: 03/01/99 at 08:59:48
From: Anonymous
Subject: Re: Logic, Groups and Identities

Thanks for the information, but I do not understand what identity 
means. Could you give me an example relating to numbers, e.g: Z+ ?

Thanks,
Anna

Date: 03/01/99 at 18:53:51
From: Doctor Schwa
Subject: Re: Logic, Groups and Identities

Identity just means the thing that leaves you unchanged. So with 
numbers, for addition, it is 0, and for multiplication it is 1.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/