I want to prove or disprove (A-B)'=B'-A'. So, let x be an
element of (A-B)' Then x is not an element of A-B This is where
I get confused... Do I say x is an element of A', x is an
element of -B'? these are not proofs that the claim is false,
however, to disprove something, you must give a counter example
To have that (A-B)' IS equal to B'-A' THIS must hold for all A
AND B. However if this equality does not hold then there must be
an A or B or both such that : (A-B)' IS not equal to B'-A' AND
Plato already gave you an example. Perhaps an example not using
Nos is the following: Let B= Φ ( the empty set), and if
substitute Φ ΙΝ (Α-Β)' we have: ( A-Φ)'= (A)'= A' BECAUSE
A-Φ=Α.Αnd substituting Φ in B'-A' we have: Φ'-A' = U- A' =A
BECAUSE Φ'=U WHERE U is the set of which A, B are subsets. Hence
we have A'= A NOT true. But we can give a proof in propositional
calculus without using a counter example by the following
procedure: Let xε( A-B)' <===> ~xεΑ v xεB as it was shown Let
xε(B'-A') <===> ~xεB & ~xεA' <====> ~xε