Question Number 216734 by mnjuly1970 last updated on 17/Feb/25
Answered by sniper237 last updated on 18/Feb/25
$${Let}\:{E}=\left\{{f}:{A}\rightarrow{B}\right\}\:\:,\:\mid{E}\mid=\mathrm{3}^{{m}} \: \\ $$$${E}\backslash{S}=\left\{{f}\in{E},\:\:\mid{f}\left({A}\right)\mid=\mathrm{1}\:{or}\:\mathrm{2}\right\} \\ $$$$\mid{E}\backslash{S}\mid=\:{C}_{\mathrm{3}} ^{\mathrm{1}} .\mathrm{1}^{{m}} +{C}_{\mathrm{3}} ^{\mathrm{2}} .\left(\mathrm{2}^{{m}} −\mathrm{1}^{{m}} −\mathrm{1}^{{m}} \right)=\mathrm{3}.\mathrm{2}^{{m}} −\mathrm{3} \\ $$$${So}\:\:\mid{S}\mid=\mathrm{3}^{{m}} −\left(\mathrm{3}.\mathrm{2}^{{m}} −\mathrm{3}\right) \\ $$
Commented by mehdee7396 last updated on 18/Feb/25
$${note}\::\:\mathrm{3}^{{m}} −\left(\mathrm{3}×\mathrm{2}^{{m}} −\mathrm{3}\right)\neq\mathrm{3}^{{m}} −\mathrm{3}×\mathrm{2}^{{m}} −\mathrm{3} \\ $$
Commented by mnjuly1970 last updated on 18/Feb/25
$$\:{thanks}\:{alot}… \\ $$
Answered by mehdee7396 last updated on 19/Feb/25
$${let}\:\:{A}=\left\{{a}_{{i}} :\:\mathrm{1}\leqslant{i}\leqslant{m}\right\}\:\&\:\:{B}=\left\{{b}_{\mathrm{1}} ,{b}_{\mathrm{2}} ,{b}_{\mathrm{3}} \right\} \\ $$$${F}=\left\{{f}:{A}\rightarrow{B}\mid{f}\left({a}\right)={b}\:\:;{a}\in{A}\:,\:{b}\in{B}\right\} \\ $$$${f}_{\mathrm{1}} =\left\{{f}:{A}\rightarrow{B}\mid{f}\left({a}_{{i}} \right)\neq{b}_{\mathrm{1}} \right\} \\ $$$${f}_{\mathrm{2}} =\left\{{f}:{A}\rightarrow{B}\mid{f}\left({a}_{\mathrm{2}} \right)\neq{b}_{\mathrm{2}} \right\} \\ $$$${f}_{\mathrm{3}} =\left\{{f}:{A}\rightarrow{B}\mid{f}\left({a}_{{i}} \right)\neq{b}_{\mathrm{3}} \right\} \\ $$$$\Rightarrow\mid{F}\mid=\mathrm{3}^{{m}} \:\:\&\:\:\mid{f}_{{i}} \mid=\mathrm{2}^{{m}} \:\&\:\:\mid{f}_{{i}} \cap{f}_{{j}} \mid=\mathrm{1}\:\:\&\:\:\mid{f}_{\mathrm{1}} \cap{f}_{\mathrm{2}} \cap{f}_{\mathrm{3}} \mid=\mathrm{0} \\ $$$$\Rightarrow\mid{S}\mid=\mid{F}\mid−\Sigma\mid{f}_{{i}} \mid+\Sigma\mid{f}_{{i}} \cap{f}_{{j}} \mid−\mid{f}_{\mathrm{1}} \cap{f}_{\mathrm{2}} \cap{f}_{\mathrm{3}} \mid \\ $$$$=\mathrm{3}^{{m}} −\mathrm{3}×\mathrm{2}^{{m}} +\mathrm{3} \\ $$$$ \\ $$