Question Number 224614 by necx122 last updated on 21/Sep/25
$${A}\:{binary}\:{operation}\:\ast\:{is}\:{defined}\:{on}\:{a}\:{set} \\ $$$${real}\:{numbers},\:{R}\:{by} \\ $$$${x}\ast{y}\:=\:\mathrm{2}{x}\:+\:\mathrm{2}{y}\:−\frac{{xy}}{\mathrm{3}}\:. \\ $$$${find}: \\ $$$$\left({i}\right)\:{The}\:{inverse}\:{of}\:{x}\:{under}\:{the}\:{operation}\:\ast \\ $$$$\left({ii}\right)\:{Truth}\:{set}\:{when}\:{m}\ast\mathrm{7}=−\mathrm{2}\ast{m} \\ $$
Answered by hfdghcdfj last updated on 22/Sep/25
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Answered by Ghisom_ last updated on 22/Sep/25
$$\mathrm{for}\:\mathrm{an}\:\mathrm{inverse}\:\mathrm{element}\:\mathrm{we}\:\mathrm{need}\:\mathrm{a}\:\mathrm{neutral} \\ $$$$\mathrm{element}\:{n} \\ $$$$\mathrm{in}\:\mathrm{our}\:\mathrm{case}\:{x}\ast{y}={y}\ast{x},\:\mathrm{so}\:\mathrm{we}'\mathrm{re}\:\mathrm{looking} \\ $$$$\mathrm{for}\:{n}\:\mathrm{with} \\ $$$$\forall{x}:\:{x}\ast{n}={x}\:\Rightarrow\:{n}\:\mathrm{must}\:\mathrm{be}\:\mathrm{independent}\:\mathrm{of}\:{x} \\ $$$${x}\ast{n}=\mathrm{2}{x}+\mathrm{2}{n}−\frac{{xn}}{\mathrm{3}}={x}\:\Rightarrow\:{n}=\frac{\mathrm{3}{x}}{{x}+\mathrm{6}} \\ $$$$\Rightarrow\:{n}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{exist}\:\Rightarrow\:\mathrm{no}\:\mathrm{inverse}\:\mathrm{exists} \\ $$$$ \\ $$$${m}\ast\mathrm{7}=−\mathrm{2}\ast{m} \\ $$$$\mathrm{the}\:\mathrm{rhs}\:\mathrm{is}\:\mathrm{not}\:\mathrm{defined} \\ $$$$−\mathrm{2}\ast{m}=−\left(\mathrm{2}\ast{m}\right)\:\mathrm{or}\:−\mathrm{2}\ast{m}=\left(−\mathrm{2}\right)\ast{m} \\ $$$$\mathrm{we}\:\mathrm{cannot}\:\mathrm{solve}\:\mathrm{this}\:\mathrm{without}\:\mathrm{further} \\ $$$$\mathrm{information} \\ $$
Commented by necx122 last updated on 22/Sep/25
$${Thank}\:{you},\:{sir} \\ $$