Let * be the binary operation on the rational number given by a*b=a+b+ab. Which of the following property does not exist for the group?(a) closure property(b) identity property(c) symmetric property(d) associative propertyI have been asked this question in class test.Question is from Groups in section Groups of Discrete Mathematics

Let * be the binary operation on the rational number given by a*b=a+b+

1.	Let * be the binary operation on the rational number given by a*b=a+b+ab. Which of the following property does not exist for the group?(a) closure property(b) identity property(c) symmetric property(d) associative propertyI have been asked this question in class test.Question is from Groups in section Groups of Discrete Mathematics
Answer» RIGHT answer is (b) identity property The best EXPLANATION: For identity e, a+e=e+a=e, ae = a+e+ae = a => e=0 and e+a = e+a+ea = a => e=0. So e=0 will be identity, for e to be identity, ae = a ⇒ a+e+ae = a ⇒ e+ae = 0 and e(1+a) = 0 which GIVES e=0 or a=-1. So, when a = -1, no identity element exist as e can be any VALUE in that case.

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