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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).

$${What}\:{is}\:{the}\:{set}\:\mathbb{Z}_{\mathrm{8}} −\left\{\mathrm{0}\right\}?\:{I}\:{met} \\ $$$${this}\:{notation}\:{in}\:{a}\:{question}\:{asking} \\ $$$${whether}\:{or}\:{not}\:{the}\:{set}\:\mathbb{Z}_{\mathrm{8}} −\left\{\mathrm{0}\right\} \\ $$$${forms}\:{a}\:{group}\:{under}\: \\ $$$${multiplication}\:\left({mod}\:\mathrm{8}\right). \\ $$

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)

$$\mathrm{i}\:\mathrm{think}\:\mathrm{that}\:\mathrm{it}\:\mathrm{could}\:\mathrm{be}\:\mathrm{the}\:\mathrm{possible} \\ $$$$\mathrm{remainder}\:\mathrm{of}\:\mathrm{a}\:\mathrm{number}\:\mathrm{by}\:\mathrm{a}\:\mathrm{divison} \\ $$$$\mathrm{with}\:\mathrm{8} \\ $$$$\mathrm{then}\:\mathrm{excluding}\:\mathrm{0} \\ $$$$\mathbb{Z}_{\mathrm{8}} =\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{7}\right\} \\ $$$$\mathbb{Z}_{\mathrm{8}} −\left\{\mathrm{0}\right\}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{7}\right\} \\ $$$$\mathrm{for}\:\mathrm{group},\:\mathrm{we}\:\mathrm{have}\:\mathrm{that}\:\mathrm{its}\:\mathrm{not}\:\mathrm{because} \\ $$$$\left\{\mathbb{Z}_{\mathrm{8}} −\left\{\mathrm{0}\right\},×\right\} \\ $$$$\mathrm{is}\:\mathrm{not}\:\mathrm{closed}\:\mathrm{under}\:\mathrm{multiplication}\:\left(\mathrm{i}\:\mathrm{cound}\:\mathrm{be}\:\mathrm{wrong}\right) \\ $$$$\mathrm{we}\:\mathrm{can}\:\mathrm{see}\:\mathrm{this}\:\mathrm{by}\:\mathrm{seting} \\ $$$${x}=\mathrm{2},{y}=\mathrm{4} \\ $$$${xy}\equiv\mathrm{2}×\mathrm{4}\equiv\mathrm{8}\equiv\mathrm{0}\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$

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} .

$${That}\:{is}\:{correct}.\:{Closure}\:{is}\:{not} \\ $$$${observed}\:{for}\:\exists{a},{b}\in\mathbb{Z}_{\mathrm{8}} −\left\{\mathrm{0}\right\}\:{under} \\ $$$${multiplication}\:\left({mod}\:\mathrm{8}\right)\:{such}\:{that} \\ $$$${a}\ast{b}\notin\mathbb{Z}_{\mathrm{8}} −\left\{\mathrm{0}\right\}\:. \\ $$

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