Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSets3 Marks
Question
For any two sets A and B, show that the following statements are equevalent: $\text{A}\subset\text{B}.$
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Answer
In order to show that the following four statements are equivalent, we need to show that (1) ⇒ (2), (2) ⇒ (3), (3) ⇒ (4) and (4) ⇒ (1) We first show that (1) ⇒ (2) We assume that $\text{A}\subset\text{B},$ and use this to show that $\text{A - B} =\phi$ Now $\text{A}-\text{B}=\{\text{x}\in\text{A : x}\not\in\text{B}\},\text{As A}\subset\text{B},$ $\therefore$ Each element of A is an element of B, $\therefore\text{A - B}=\phi$ Hence, we have proved that (1) ⇒ (2).
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