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Question Number 1171 by 112358 last updated on 09/Jul/15
What is the set Z_8 −{0}? I met  this notation in a question asking  whether or not the set Z_8 −{0}  forms a group under   multiplication (mod 8).
WhatisthesetZ8{0}?ImetthisnotationinaquestionaskingwhetherornotthesetZ8{0}formsagroupundermultiplication(mod8).
Answered by 123456 last updated on 10/Jul/15
i think that it could be the possible  remainder of a number by a divison  with 8  then excluding 0  Z_8 ={0,1,2,3,4,5,6,7}  Z_8 −{0}={1,2,3,4,5,6,7}  for group, we have that its not because  {Z_8 −{0},×}  is not closed under multiplication (i cound be wrong)  we can see this by seting  x=2,y=4  xy≡2×4≡8≡0(mod 8)
ithinkthatitcouldbethepossibleremainderofanumberbyadivisonwith8thenexcluding0Z8={0,1,2,3,4,5,6,7}Z8{0}={1,2,3,4,5,6,7}forgroup,wehavethatitsnotbecause{Z8{0},×}isnotclosedundermultiplication(icoundbewrong)wecanseethisbysetingx=2,y=4xy2×480(mod8)
Commented by 112358 last updated on 10/Jul/15
That is correct. Closure is not  observed for ∃a,b∈Z_8 −{0} under  multiplication (mod 8) such that  a∗b∉Z_8 −{0} .
Thatiscorrect.Closureisnotobservedfora,bZ8{0}undermultiplication(mod8)suchthatabZ8{0}.

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