Question
For any two sets A and B, prove that.
$\text{A}\subset\text{B}\Rightarrow\text{A}\cap\text{B}=\text{A}.$
$\text{A}\subset\text{B}\Rightarrow\text{A}\cap\text{B}=\text{A}.$
✓
Answer
Let $\text{x}\in\text{A}\subset\text{B}.$ Then
$\Rightarrow\text{x}\in\text{B}$
Let and $\text{x}\in\text{A}\cap\text{B}$
$\Leftrightarrow\text{x}\in\text{A and x}\in\text{B}$
$\Leftrightarrow\text{x}\in\text{A and x}\in\text{A}$ $(\because\text{A}\subset\text{B})$
$\therefore(\text{A}\cap\text{B})=\text{A}.$
$\Rightarrow\text{x}\in\text{B}$
Let and $\text{x}\in\text{A}\cap\text{B}$
$\Leftrightarrow\text{x}\in\text{A and x}\in\text{B}$
$\Leftrightarrow\text{x}\in\text{A and x}\in\text{A}$ $(\because\text{A}\subset\text{B})$
$\therefore(\text{A}\cap\text{B})=\text{A}.$
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