Are the following proofs correct??
a) AxB =Φ <====> [ (a,b)εAxB <-----> (a,b)εΦ] <=====>[(aεA & bεB)<------>(aεΦ & bεΦ)] <====>[(aεA----->aεΦ) & (bεB------>bεΦ)] ======>(aεA----->aεΦ)====> A= Φ ====> A=Φ v B=Φ.
Ηence:..... AxB =Φ =====> A=Φ v B=Φ
b) AxB =Φ <====> [~(a,b)εΑxB] <=====>
[ ~aεA v ~bεB] <========>
( ~aεA) v (~bεB) <======>
( ~aεA) v (~bεB) <======>
A= Φ v B=Φ ........justify your answer,if possible
Where:.........AxB ={ (a,b): aεA & bεB } , Φ is the empty set and " ~" means not.
Thanks