quiz9 - if a and b are in S then so is ab Let T and U be...

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Putnam Exam Seminar Quiz 9 Fall 2010 November 15 Problem 1. Consider a set S with a binary operation * , that is, for each a,b S , a * b S . Assume that ( a * b ) * a = b for all a,b S . Prove that a * ( b * a ) = b for all a,b S . [Putnam Exam, 2001, A1] Problem 2. Let S be a non-empty set with an associative operation that is left and right cancellative ( xy = xz implies y = z , and yx = zx implies y = z ). Assume that for every a in S the set { a n : n = 1 , 2 , 3 ,... } is ﬁnite. Must S be a group? [Putnam Exam, 1989, B2] Problem 3. Let S be a set of real numbers which is closed under multiplication (that is,
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Unformatted text preview: if a and b are in S , then so is ab ). Let T and U be disjoint subsets of S whose union is S . Given that the product of any three (not necessarily distinct) elements of T is in T and that the product of any three elements of U is in U , show that at least one of the two subsets T,U is closed under multiplication. [Putnam Exam, 1995, A1]...
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This note was uploaded on 02/29/2012 for the course MATH 1190 taught by Professor Ryandaileda during the Fall '10 term at Trinity University.

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