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# sols16 - 16. OCTOBER 25TH: INDUCTION. 39 16. October 25th:...

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16. OCTOBER 25TH: INDUCTION. 39 16. October 25th: Induction. 16.1 . An order relation ° on a set S is a ‘total’ ordering if for every two elements a and b in S ,e ither a ° b or b ° a . Give an example of an order relation that is not total. Give an example of a total ordering that is not a well ordering. Prove that every well-ordering is total. Answer: Consider the relation ° on R × R deFned by prescribing that ( x 1 ,y 1 ) ° ( x 2 ,y 2 )i f ( x 1 x 2 ) ( y 1 y 2 ) . This is an order relation (right?) and it is not a total ordering, since for example (0 , 1) ±° (1 , 0) and (1 , 0) ±° (0 , 1). The order relation on R 0 is not a well-ordering (Example 7.1), and it is ato ta lo rde r ing : g ivenanytwononnega t iverea lnumbe r s r 1 , r 2 ,e i the r r 1 r 2 or r 2 r 1 . Let ° be a well-ordering on a set S ,andlet a, b S .I f a = b ,theninpart icu lar a ° b ,andthe reisno th ingtop rove . I f a ± = b ,thencons ide rthesubse t T = { a, b } of S .S ince ° is a well-ordering, T has a minimum. If a is the minimum, then a ° b . If b is the minimum, then b ° a .Thu sn e c e s s a r i ly a ° b or b ° a ,wh ichshowsthat ° is a total ordering. ° 16.2 . Let ° be a well-ordering on a set S .P r o

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## This note was uploaded on 12/14/2011 for the course MGF 3301 taught by Professor Aluffi during the Fall '11 term at FSU.

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sols16 - 16. OCTOBER 25TH: INDUCTION. 39 16. October 25th:...

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