tag:blogger.com,1999:blog-12604614.post114162223019789059..comments2023-06-12T05:26:38.585-07:00Comments on Math Refresher: Dedekind CutLarry Freemanhttp://www.blogger.com/profile/06906614246430481533noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-12604614.post-83781217695629169732010-01-06T08:47:07.891-08:002010-01-06T08:47:07.891-08:00Hi Anonymous,
I agree with your criticisms of the...Hi Anonymous,<br /><br />I agree with your criticisms of the definitions. I'll update the definitions and repost.<br /><br />Thanks very much for your comments!<br /><br />-LarryLarry Freemanhttps://www.blogger.com/profile/06906614246430481533noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-24882832805300556722010-01-05T06:10:03.867-08:002010-01-05T06:10:03.867-08:00Then definition 1 makes no sense as you say a dede...Then definition 1 makes no sense as you say a dedekind cut is a subset of intergers which are less than that dedekind cut. You can't use a dedekind cut to define a dedekind cut. You also haven't defined what less than means as these are sets you are dealing with not numbers. a < b should be defined as a is a subset of b, but you can't do this until you define dedekind cuts.<br />The problem with your definition of multiplication is that you say a*b is the set of all products of all rational numbers in the original cuts a and b. So if we multiply 2 and 1 we can take -1 which is in the cut 1 and -100 which is in the cut 2. Then 100=(-1)*(-100) must be in the cut 2*1.<br />Your definiton of division is also problematic as you only specify that b can not 0 for the cut a/b. but if b>0 then 0 will be in the cut b and so x/0 for x in cut a will be in a/b. But this makes no sense.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-12604614.post-73729702939388990892010-01-04T09:31:53.088-08:002010-01-04T09:31:53.088-08:00I'm not clear on your question.
Definition 1 ...I'm not clear on your question.<br /><br />Definition 1 defines a Dedekind cut in terms of rational numbers.<br /><br />Definition 2 defines real numbers in terms of Dedekind cut.<br /><br />100 = (-1)*(100) so I'm not sure what you are saying here. If you add more details, I'll be glad to respond.Larry Freemanhttps://www.blogger.com/profile/06906614246430481533noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-37168844070767663782010-01-04T08:40:01.222-08:002010-01-04T08:40:01.222-08:00You seem to have a circular arguement here. In def...You seem to have a circular arguement here. In definition 1 you define a dedekind cut by a real number but you can't do this as the real numbers are supposed to be defined by dedekind cuts. You need to define dedekind cuts without real numbers and then define real numbers as dedekind cuts.<br />Also your definition of multiplication is wrong as it allows 100 to be in the set 1*2 as 100=(-1)*(-100).Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-12604614.post-22081975472974495572010-01-01T03:40:08.250-08:002010-01-01T03:40:08.250-08:00That's Definition 5 and Definition 6.
The A +...That's Definition 5 and Definition 6.<br /><br />The A + B set is a new set which consists of every combination of a + b where a is an element of A and b is an element of B.<br /><br />A - B set is a new set which consists of every combination of a - b where a is an element of A and b is an element of B.<br /><br />-LarryLarry Freemanhttps://www.blogger.com/profile/06906614246430481533noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-54993070685232029462010-01-01T02:22:30.824-08:002010-01-01T02:22:30.824-08:00What would + and - mean exactly? It isn't clea...What would + and - mean exactly? It isn't clear to me. <br />Thanks.Haroonhttps://www.blogger.com/profile/17573502258888413569noreply@blogger.com