For any two sets A and B, show that the following statements are … - Vidyadip

Question

For any two sets A and B, show that the following statements are equevalent:
$\text{A}\subset\text{B}.$

✓

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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