Consider the operation for real numbers x and y defined by x y D xy=.x C y/. 2 3 D 2

Chapter 40, Problem 40.3

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Consider the operation ? for real numbers x and y defined by x ? y D xy=.x C y/. 2 ? 3 D 2 3=.2 C 3/ D 6 5 . We note that .R; ?/ is not a group for a variety of reasons, not the least of which is that x ? y might not be defined (we might divide by 0). The situation, however, is not hopeless; lets do some repairs. First, lets deal with the divide-by-zero issue by extending the real numbers to also include the number 1. With this extension, we can have .3/ ? 3 D .3/ 3=.3 C 3/ D 9=0 D 1. Thats acceptable, but 0=0 is a worse problem, so lets simply ban 0 from the set of allowable values for ?. That is, we define R D R f0g [ f1g: That is R consists of all nonzero real numbers and the additional number 1. Give sensible meanings to x ? 1, 1 ? x, and 1 ? 1 (where x is a nonzero real number) and show that .R ; ?/ is an Abelian group.Let .G; / be a group with G D fa; b; cg. Here is an incomplete operation table for : a b ca a b cb ? ? ?c ? ? ?Find the missing entries.