For all sets A, B and C, if A , then A - Vidyadip

Question

For all sets A, B and C, if $\text{A}\subset\text{B},$ then $\text{A}\cup\text{C}\subset\text{B}\cup\text{C}$

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Answer

Suppose $\text{A}\subset\text{B}$
Let $\text{x}\in\text{A}\cup\text{C}$
$\Rightarrow \text{x}\in \text{A}$ or $\text{x}\in\text{C}$
$\Rightarrow \text{x}\in\text{B}$ and $\text{x}\in\text{C}\ \big[\because \text{A}\subset\text{B}\big]$
$\Rightarrow \text{x}\in(\text{B}\cup\text{C})$
$\Rightarrow (\text{A}\cup\text{C})\subset(\text{B}\cup\text{C})$
Hence, the given statement is 'True'.

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