Prove that the sets $A$ and $B , A - B = B ^{\prime}- A ^{\prime}$ as true.
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Answer
Here we are to prove, $\begin{array}{l}(A-B) \subseteq\left(B^{\prime}-A^{\prime}\right) \text { and }\left(B^{\prime}-A^{\prime}\right) \subseteq(A-B) \\\text { Let } \quad x \in(A-B) \Rightarrow x \in A \text { and } x \notin B \\\Rightarrow x \notin A^{\prime} \text { and } x \in B^{\prime} \\\Rightarrow x \in B^{\prime} \text { and } x \notin A^{\prime} \\\Rightarrow x \in\left(B^{\prime}-A^{\prime}\right) \\\therefore \quad(A-B) \subseteq\left(B^{\prime}-A^{\prime}\right)\ldots\ldots (1)\end{array}$ $ \begin{aligned}\text {Again}\quad x \in\left(B^{\prime}- A ^{\prime}\right. & \Rightarrow x \in B^{\prime} \text { and } x \notin A^{\prime} \\ & \Rightarrow x \notin B \text { and } x \in A \\ & \Rightarrow x \in A \text { and } x \notin B \\ & \Rightarrow x \in(A- B )\end{aligned}$ $ \therefore \quad\left(B^{\prime}-A^{\prime}\right) \subseteq(A-B)\ldots\ldots (2)$ From equations (i) and (ii), $A - B = B ^{\prime}- A ^{\prime}$ is true.
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