إذا كانت المجموعات:

$\begin{array}{rl}& A=\{-3,-2,-1,0,1,2,3,4\},\\ & B=\{-3,-1,1,3,5,6\},\\ & C=\{-2,1,3,4,7,8\}\end{array}$

فأثبت ما يأتي:

$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$

$$\begin{gathered} A\cap (B\cup C)=\{-3,-2,-1,0,1,2,3,4\}\cap \{-3,-1,1,3,5,6\} \\ =\{-3,-1,1,3\} \\ (A\cap B)\cup (A\cap C)=\{-3,-1,1,3\}\cup \{1,3\} \\ =\{-3,-1,1,3\} \\ \therefore A\cap (B\cup C)=(A\cap B)\cup (A\cap C) \end{gathered}$$

فكر واكتب

إذا كانت المجموعات:

$\begin{array}{rl}& A=\{-3,-2,-1,0,1,2,3,4\},\\ & B=\{-3,-1,1,3,5,6\},\\ & C=\{-2,1,3,4,7,8\}\end{array}$

فأثبت ما يأتي:

$\mathrm{A}\cap \mathrm{B}=\mathrm{B}\cap \mathrm{A}$

$$\begin{gathered} \text{A}\cap B=\{-3,-1,1,3\} \\ B\cap A=\{3,1,-1,-3\} \\ \therefore \mathrm{A}\cap \mathrm{B}=\mathrm{B}\cap \mathrm{A} \end{gathered}$$

$\mathrm{A}\cap \mathrm{C}=\mathrm{C}\cap \mathrm{A}$

$$\begin{gathered} A\cap C=\{1,3\},C\cap A=\{1,3\} \\ \therefore A\cap C=C\cap A \end{gathered}$$

$\mathrm{A}\cup \mathrm{B}=\mathrm{B}\cup \mathrm{A}$

$$\begin{gathered} \begin{array}{rl}& \mathrm{A}\cup \mathrm{B}=\{-3,-2,-1,0,1,2,3,4,5,6\}\\ & \mathrm{B}\cup \mathrm{A}=\{-3,-2,-1,0,1,2,3,4,5,6\}\end{array} \\ \therefore \mathrm{A}\cup \mathrm{B}=\mathrm{B}\cup \mathrm{A} \end{gathered}$$

$A\cup C=C\cup A$

$$\begin{gathered} \begin{array}{rl}& A\cup C=\{-3,-2,-1,0,1,2,3,4,7,8\}\\ & C\cup A=\{-3,-2,-1,0,1,2,3,4,7,8\}\end{array} \\ \therefore A\cup C=C\cup A \end{gathered}$$

$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$

$$\begin{gathered} A\cap (B\cup C)=\{-3,-2,-1,0,1,2,3,4\}\cap \{-3,-1,1,3,5,6\} \\ =\{-3,-1,1,3\} \\ (A\cap B)\cup (A\cap C)=\{-3,-1,1,3\}\cup \{1,3\} \\ =\{-3,-1,1,3\} \\ \therefore A\cap (B\cup C)=(A\cap B)\cup (A\cap C) \end{gathered}$$

اكتب: عناصر مجموعة تقاطع المجموعتين:

$\{x\in Z:-5<x<1\}\cap \{x\in Z:-2<x<6\}$

$\{-4,-3,-2,-1,0\}\cap \{-1,0,1,2,3,4,5\}=\{-1,0\}$