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A binaryoperation * on the set {0,1,2,3,4,5} is defined as:`a*b={a+b a+b-6" if "a+b |
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Answer» Let `X = {0, 1, 2, 3, 4, 5}.` The given operation `**` on `X` is defined as: `a**b={a+b if a+b<6` `a+b-6 if a+b≥6}` An element `einX` is the identity element for the operation `**`, if`a**e=a=e**a` for `a in X`. For `a in X`, `a**0=a+0=a` `0**a=0+a=a ` `:. a**0=a=0**a` for `a in X`. Thus, `0` is the identity element for the given operation `**`. An element `a in X` is invertible if there exists `b in X` such that `a**b=e=b**a`. `=>a**b ={(a+b=0=b+a,if a+b<6` `a+b−6=0=b+a−6if a+b≥6)}` `=>a=−b or b=6−a` But, `X = {0, 1, 2, 3, 4, 5}` and `a,b in X`.Then, `a!=−b`. Therefore, `b=6−a` is the inverse of `a`. Hence, the inverse of an element `a in X,a != 0` is `6−a `. |
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