For three sets A, B and C, show that. A -B -A. - Vidyadip

Question

For three sets A, B and C, show that.
$\text{A}\subset\text{B}\Rightarrow\text{C}-\text{B}\subset\text{C}-\text{A}.$

✓

Answer

Given $\text{A}\subset\text{B}$
To show: $\text{C} - \text{B }\subset\text{ C} - \text{A}$
Let $\text{x}\in\text{C}-\text{B}$
$\Rightarrow \text{x}\in\text{C and x}\not\in\text{B}$ [by defination of C - B]
$\Rightarrow \text{x}\in\text{C and x}\not\in\text{A}$ $[\because\text{A}\subset\text{B}]$
This can be seen by the venn diagram above
$\Rightarrow \text{x}\in\text{C - A}$ [by defination of C - A]
Thus $\text{x}\in\text{C - B}\Rightarrow\text{x}\in\text{C - A.}$ Thus is true for all $\text{x}\in\text{C}-\text{B}$
$\therefore\text{C - B}\subset\text{C - A}.$

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