Question

If A is any set , prove that :

$A⊈\varphi \iff A=\varphi$.

Answer 1

The symbol ${}^{\prime }{\iff }^{\prime }$ stands for if and only if (in short if).

In order to show that two sets A and B are equal we show that $A⊈BandB⊈A$.

We have $A⊈\varphi$, $\because$ is a subset of every set.

$\therefore \varphi ⊈A$

Hence A = $\varphi$

To show the backward implication, suppose that A = $\varphi$

$\because$ every set is a subset of itself

$\therefore \varphi =A\subseteq \varphi$.

Hence, proved.