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Show that the operation * on Z define by a* b =a+b+1 for all a,b in Z Satisfies (i) the closure property (ii) the associative law and (iii) the commutative law ltrbgt (iv) find the identity element inZ (v) what is the inverse iof an element a in Z ? |
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Answer» Solution :(i) closure porperty LET a `in` A ,b in Z then a*b=a+b+1 Nowa `in` Z , b in Z `RARR` a+b in Z `rarr` a+b +1 `in` Z `therefore` * on Z satisfies the closure propety (ii) Associative law For all a,b,c `in` Z we have (a*b)*c=(a+b+1)*c =(a+b+1)+c+1 =a+b+c+2 a*(b*c)=a*(b+c+1) =a+(b+c+1)+1 =a+b+c+2 `therefore` (a*b)*c=a*(b*c) (iii) commutative law For all a,b `in` Z we have a*b=a+b+1 ltbrgeb+a+1 [`therefore` a+b+=b+1] =b*a (iv) Existence of identity element Let e be the identity element in Z Then a * e =a `rarr` a+e+1=a `rarr` =-1 Thus -1 in Z is the identity element for * (V) Existence of inverser Let a in Z and let ISTS iverse e be b then a*b=-1 `rarr` a+b+1=-1 `rarr` b =-(2+a) clearly 2`in` Z a in Z `rarr` -(2+a) `in` Z Thus each a in Z has -(2+a)in Z as its INVERSE |
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