if we are to prove
X-(X-A) = A,
what we normally do is for an arbitrary x suppose ,
x E X-(X-A)
--> proof
--> x E A
and then
x E A
--->proof
---> x E X-(X-A)
this tells us X-(X-A) = A,
its perfect and theres no wrong with it.
but my question is what happens if I use biconditional statement as
x E X-(X-A) <---> x E X AND x ~E X -A
<---> some proof here
<---> x E A
then we dont want to split it into two parts as did earlier.
but why dont we normally prove like this???
suppose i have to prove X-(X-A) = A,
1st step :and 2nd step:for an arbitrary x suppose ,
x E X-(X-A)
--> proof goes here
--> x E A
this is the normal structure of proving this sum.ok?x E A
--->proof goes here
---> x E X-(X-A)
but if i do it like this...using the definition of set difference,
is it wrong?x E X-(X-A) <---> x E X AND x ~E X -A
<---> some proof here
<---> x E A