Question-187186

Question Number 187186 by 073 last updated on 14/Feb/23

Commented by 073 last updated on 14/Feb/23

$$\mathrm{please}\:\mathrm{solution}?? \\ $$

Commented by Rasheed.Sindhi last updated on 15/Feb/23

$$\mathrm{k}=\mathrm{4}\:\left(\mathrm{please}\:\mathrm{confirm}\:\mathrm{the}\:\mathrm{answer}\right) \\ $$

Commented by mr W last updated on 15/Feb/23

$${k}=\mathrm{4}\:{is}\:{correct}. \\ $$

Commented by Rasheed.Sindhi last updated on 16/Feb/23

$$\boldsymbol{{sir}}\:{mr}\:{W},\:{when}\:{you}\:{have}\:{some}\:{leisure} \\ $$$${please}\:{see}\:{my}\:{answer}\:{to}\:{Q}#\mathrm{186966} \\ $$$${for}\:{its}\:{health}! \\ $$

Commented by 073 last updated on 16/Feb/23

$$\mathrm{thanks} \\ $$

Commented by 073 last updated on 16/Feb/23

$$\mathrm{thanks} \\ $$

Answered by Rasheed.Sindhi last updated on 15/Feb/23

$$\begin{array}{|c|c|c|c|c|c|}{\bigstar}&\hline{\mathrm{1}}&\hline{\mathrm{2}}&\hline{\mathrm{3}}&\hline{\mathrm{4}}&\hline{\mathrm{5}}\\{\mathrm{1}}&\hline{\mathrm{5}}&\hline{\mathrm{1}}&\hline{\mathrm{2}}&\hline{\mathrm{3}}&\hline{\mathrm{4}}\\{\mathrm{2}}&\hline{\mathrm{1}}&\hline{\mathrm{2}}&\hline{\mathrm{3}}&\hline{\mathrm{4}}&\hline{\mathrm{5}}\\{\mathrm{3}}&\hline{\mathrm{2}}&\hline{\mathrm{3}}&\hline{\mathrm{4}}&\hline{\mathrm{5}}&\hline{\mathrm{1}}\\{\mathrm{4}}&\hline{\mathrm{3}}&\hline{\mathrm{4}}&\hline{\mathrm{5}}&\hline{\mathrm{1}}&\hline{\mathrm{2}}\\{\mathrm{5}}&\hline{\mathrm{4}}&\hline{\mathrm{5}}&\hline{\mathrm{1}}&\hline{\mathrm{2}}&\hline{\mathrm{3}}\\\hline\end{array} \\ $$$$\mathrm{A}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5}\right\} \\ $$$$\mathrm{x}\bigstar\mathrm{e}=\mathrm{x}\:\forall\mathrm{x}\in\mathrm{A}\Rightarrow\mathrm{e}=\mathrm{2} \\ $$$$\mathrm{1}^{−\mathrm{1}} =\mathrm{3}\:\:\:\left[\because\:\mathrm{1}\bigstar\mathrm{3}=\mathrm{2}\right] \\ $$$$\mathrm{2}^{−\mathrm{1}} =\mathrm{2}\:\:\:\left[\because\:\mathrm{2}\bigstar\mathrm{2}=\mathrm{2}\right] \\ $$$$\mathrm{3}^{−\mathrm{1}} =\mathrm{1}\:\:\:\:\left[\because\:\mathrm{3}\bigstar\mathrm{1}=\mathrm{2}\right] \\ $$$$\mathrm{4}^{−\mathrm{1}} =\mathrm{5}\:\:\:\:\left[\because\:\mathrm{4}\bigstar\mathrm{5}=\mathrm{2}\right] \\ $$$$\mathrm{5}^{−\mathrm{1}} =\mathrm{4}\:\:\:\left[\because\:\mathrm{5}\bigstar\mathrm{4}=\mathrm{2}\right] \\ $$$$ \\ $$$$\left(\mathrm{2}\bigstar\mathrm{k}\right)^{−\mathrm{1}} \bigstar\mathrm{5}=\mathrm{3} \\ $$$$\mathrm{k}^{−\mathrm{1}} \bigstar\mathrm{5}=\mathrm{3}\:\:\:\left[\because\mathrm{2}\bigstar\mathrm{k}=\mathrm{k}\right] \\ $$$$\mathrm{k}^{−\mathrm{1}} \bigstar\mathrm{5}\bigstar\mathrm{5}^{−\mathrm{1}} =\mathrm{3}\bigstar\mathrm{5}^{−\mathrm{1}} \\ $$$$\mathrm{k}^{−\mathrm{1}} \bigstar\left(\mathrm{5}\bigstar\mathrm{5}^{−\mathrm{1}} \right)=\mathrm{3}\bigstar\mathrm{4}\:\:\left[\because\mathrm{5}^{−\mathrm{1}} =\mathrm{4}\right] \\ $$$$\mathrm{k}^{−\mathrm{1}} \bigstar\mathrm{2}=\mathrm{3}\bigstar\mathrm{4}\:\:\:\:\:\left[\because\mathrm{5}\bigstar\mathrm{5}^{−\mathrm{1}} =\mathrm{2}\right] \\ $$$$\mathrm{k}^{−\mathrm{1}} =\mathrm{5}\:\:\:\:\:\left[\because\:\mathrm{k}^{−\mathrm{1}} \bigstar\mathrm{2}=\mathrm{k}^{−\mathrm{1}} \right] \\ $$$$\mathrm{k}=\mathrm{4} \\ $$

Commented by 073 last updated on 16/Feb/23

$$\mathrm{thanks}\:\mathrm{alot}\:\mathrm{sir} \\ $$

Answered by mr W last updated on 15/Feb/23

$${x}\ast{e}={x}\:\Rightarrow{e}=\mathrm{2} \\ $$$${x}\ast{x}^{−\mathrm{1}} ={e}=\mathrm{2}\:\Rightarrow\mathrm{1}^{−\mathrm{1}} =\mathrm{3},\:\mathrm{2}^{−\mathrm{1}} =\mathrm{2},\:\mathrm{3}^{−\mathrm{1}} =\mathrm{1},\:\mathrm{4}^{−\mathrm{1}} =\mathrm{5},\:\mathrm{5}^{−\mathrm{1}} =\mathrm{4} \\ $$$$\mathrm{5}\ast\mathrm{5}=\mathrm{3}\:\Rightarrow\left(\mathrm{2}\ast{k}\right)^{−\mathrm{1}} =\mathrm{5}\:\Rightarrow\mathrm{2}\ast{k}=\mathrm{4}\:\Rightarrow{k}=\mathrm{4}\:\checkmark \\ $$

Commented by 073 last updated on 16/Feb/23

$$\mathrm{nice}\:\mathrm{solution} \\ $$

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