Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSets4 Marks
Question
For any two sets of A and B, prove that:$\text{A}'\cup\text{B}=\text{U}\Rightarrow\text{A}\subset\text{B.}$
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Answer
We have $\text{A}'\cup\text{B}=\cap,$ the universal set To show: $\text{A}\subset\text{B}$ Let, $\text{x}\in\text{A}$ $\Rightarrow\text{x}\not\in\text{A}'$ $[\because\text{A}\cap\text{A}'=\phi]$ $\because\text{x}\in\text{A and A}\subset\cup$ $\Rightarrow\text{x}\in\cup$ $\Rightarrow\text{x}\in(\text{A}'\cup\text{B})$ $[\because\cup=\text{A}'\cup\text{B}]$ But, $\text{x}\not\in\text{A}',$ $\therefore\text{ x}\in\text{B}$ Thus, $\text{x}\in\text{A}\Rightarrow\text{x}\in\text{B}$ This is true for all $\text{x}\in\text{A}$ $\therefore\text{ A}\subset\text{B.}$
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