Question

Show that the following four conditions are equivalent:

(i) $A\subset B$ (ii) $A-B=\Phi$

(ii) $A\cup B=B$ (iv) $A\cap B=A$

Answer 1

$\left(i\right)\Rightarrow \left(ii\right)$

A - B = {x : x $\in$ A and x $\notin$ B}

Since $A\subset B$

$\therefore A-B=\Phi$

$\left(ii\right)\Rightarrow \left(iii\right)$

$A-B=\Phi \Rightarrow A\subset B\Rightarrow A\cup B=B$

$\left(iii\right)\Rightarrow \left(iv\right)$

$A\cup B=B\Rightarrow A\subset B\Rightarrow A\cap B=A$

$\left(iv\right)\Rightarrow \left(i\right)$

$A\cap B=A\Rightarrow A\subset B$

Thus $\left(i\right)\iff \left(ii\right)\iff \left(iii\right)\iff \left(iv\right)$