If A is any set, prove that: A = . - Vidyadip

Question

If A is any set, prove that: $\text{A}\subseteq\phi\Leftrightarrow\text{A}=\phi.$

✓

Answer

The symbol '⇔' stands for if and only if (in short if). In order to show that 2 sets A and B are equel we show htat $\text{A}\subseteq\text{B}$ and $\text{B}\subseteq\text{A}.$
We have $\text{A}\subseteq\phi.\ \because \phi$ is a subset of every set
$\therefore\ \phi\subset\text{A}$
Hence $\text{A} =\phi$
To show the backward implication, suppose that $\text{A}= \phi$
$\because$ Every set is a subset of itself.
$\therefore\ \phi=\text{A}\subseteq \phi$
Hence, proved.

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