Idempotent Laws

Relations And Functions

Question

$space space space space space space space space space straight A space union space straight A space equals space straight A$

Solution

Let x be any arbitary element of $space space space space space space space open parentheses straight A union straight A close parentheses$
i.e. $space space space space space space space space space straight x element of open parentheses straight A union straight A close parentheses space rightwards double arrow space straight x element of straight A space space or space straight x element of straight A$
$space space space space space rightwards double arrow space space space space space space space space space space space space space space space space space space space space space space space straight x element of space straight A$

∴ $space space space space space space space space space space space space space space space straight x element of straight A left parenthesis straight A union straight A right parenthesis space space space space space space space space space rightwards double arrow space space space space space straight x element of straight A$
$space rightwards double arrow space space space space space space space space space open parentheses straight A union straight A close parentheses bottom enclose subset of space straight A$ ......(i)

Similarly we can show that $space space space space space space straight A space bottom enclose subset of space open parentheses straight A space union space straight A close parentheses$ ......(ii)
Using (i) and (ii), we get $space space space space space space space space space space space space space straight A space union space straight A$