ข้อสอบ PAT 1 - ตุลาคม 2559

$$\begin{gathered} \left(\mathrm{ก}\right) \mathrm{พิจารณาฝั่งซ้าย} (\mathrm{A}\cap \mathrm{B})\cup \mathrm{C}=\left(\mathrm{A}\cup \mathrm{C}\right)\cap \left(\mathrm{B}\cup \mathrm{C}\right) \\ \mathrm{พิจารณาฝั่งขวา} \mathrm{A}\cap \left(\mathrm{B}\cup \mathrm{C}\right)=\left(\mathrm{A}\cap \mathrm{B}\right)\cup \left(\mathrm{A}\cap \mathrm{C}\right) \\ \mathrm{ดังนั้น}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \left(\mathrm{A}\cap \mathrm{B}\right)\cup \mathrm{C}\ne \mathrm{A}\cap \left(\mathrm{B}\cup \mathrm{C}\right) \\ \mathrm{เนื่องจาก} \left(\mathrm{A}\cup \mathrm{C}\right)\cap \left(\mathrm{B}\cup \mathrm{C}\right)\ne \left(\mathrm{A}\cap \mathrm{B}\right)\cup \left(\mathrm{A}\cap \mathrm{C}\right) \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{ข้อ} \left(\mathrm{ก}\right) \mathrm{ผิด} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ข}\right) \mathrm{กำหนดให้} \mathrm{A}=\left\{0,1\right\} , \mathrm{B}=\left\{1,2\right\} \\ \mathrm{จะได้} \mathrm{P}\left(\mathrm{A}\right)=\left\{\varnothing ,\left\{0\right\},\left\{1\right\},\left\{0,1\right\}\right\} \hspace{0.17em}\hspace{0.17em}\mathrm{P}\left(\mathrm{B}\right)=\left\{\varnothing ,\left\{1\right\},\left\{2\right\},\left\{1,2\right\}\right\} \\ \mathrm{P}\left(\mathrm{A}\right)-\mathrm{P}\left(\mathrm{B}\right)=\left\{\left\{0\right\},\left\{0,1\right\}\right\} \\ \mathrm{P}\left(\mathrm{A}-\mathrm{B}\right)=\mathrm{P}\left(\left\{0\right\}\right) \\ =\left\{\varnothing ,\left\{0\right\}\right\} \end{gathered}$$

$$\begin{gathered} \mathrm{ดังนั้น} \hspace{0.17em}\hspace{0.17em} \mathrm{P}\left(\mathrm{A}\right)-\mathrm{P}\left(\mathrm{B}\right)\not\subset \mathrm{P}\left(\mathrm{A}-\mathrm{B}\right) \\ \mathrm{เนื่องจาก} \left\{\left\{0\right\},\left\{0,1\right\}\right\}\not\subset \left\{\varnothing ,\left\{0\right\}\right\} \mathrm{ข้อ} \left(\mathrm{ข}\right) \mathrm{ผิด} \\ \left(\mathrm{ค}\right) \mathrm{กำหนดให้} \varnothing \subset \mathrm{P}\left(\varnothing \right) ; \mathrm{เซตว่างเป็นสับเซตของทุกเซต} \\ \mathrm{จะได้} \mathrm{P}\left(\varnothing \right)\subset \mathrm{P}\left(\mathrm{P}\left(\varnothing \right)\right) ; \mathrm{ใส่} \mathrm{P} \mathrm{ทั้ง} 2 \mathrm{ข้าง} \\ \mathrm{ดังนั้น} \mathrm{P}\left(\mathrm{P}\left(\varnothing \right)\right)\subset \mathrm{P}\left(\mathrm{P}\left(\mathrm{P}\left(\varnothing \right)\right)\right) \mathrm{ข้อ} \left(\mathrm{ค}\right) \mathrm{ถูก} \end{gathered}$$