Question

Show that the following four conditions are equivalent:
$\left(i\right)A\subset B$
$\left(ii\right)A-B=\varphi$
$\left(iii\right)A\cup B=B$
$\left(iv\right)A\cap B=A$

Answer 1

Part $1:$ Showing condition $\left(i\right)$ is equivalent to condition $\left(ii\right).$
Let $A\subset B$
$\Rightarrow$ All elements of set $A$ are in set $B.$
So, $A$ has no element different from $B.$
$\Rightarrow A-B=\varphi$

Part $2:$ Showing condition $\left(ii\right)$ is equivalent to condition $\left(iii\right).$
$A-B=\varphi \Rightarrow A$ has no element different form $B$
So, all elements of $A$ are in $B.$
$\Rightarrow A\cup B=B$

Part $3:$ Showing condition $\left(iii\right)$ is equivalent to condition $\left(iv\right).$
$A\cup B=B$
$\Rightarrow$ All elements of $A$ are in $B.$ So, the common elements of $A$ and $B$ must be the elements of $A.$
$\therefore A\cap B=A$

Part $4:$ Showing condition $\left(iv\right)$ is equivalent to condition $\left(i\right).$
$A\cap B=A$
$\Rightarrow A$ is the smaller set and all the elements of $A$ are in $B$ also.
$\therefore A\cap B=A\Rightarrow A\subset B$

Thus , $\left(i\right)\leftrightarrow \left(ii\right)\leftrightarrow \left(iii\right)\leftrightarrow \left(iv\right)\leftrightarrow \left(i\right)$