Logic, Groups, and IdentitiesDate: 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/ |
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