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:
$\text{A}\cap\text{B}=\text{A}.$

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Answer

Finally we show that (4) ⇒ (1), which will prove the equqvalence of the 4 statement.
So, assume that $\text{A}\cap\text{B}=\text{A}$
To show: $\text{A}\subset\text{B}$
$\because\text{A}\cap\text{B}=\text{A}$ therefore $\text{A}\subset\text{B},$ and so (4) ⇒ (1) is true.
Hence, (1) ⇔ (2) ⇔ (3) ⇔ (4).

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