I was understanding this proof :
(A-B) X C = (A X B) - (B X C)
A transition in statements in proof seemed so wrong to me , which are following :
$(x \in A \;and\; x \notin B) \;and \; y \in C$
$(x \in A\; and \;y \in C) \;and\; (x \notin B\; and \;y \in C)$
$(x,y) \in (A X C) \;and\; (x,y) \notin (B X C)$
my question lies in last two statements , how can we write $(x\notin B\;and\;y \in C)$ to $(x,y) \notin (B X C)$ ?
I mean statement $(x \notin B \;and\; y \in C)$ says x
doesn't belongs to B
and y
belongs to C
.
It's equivalent (in proof) statement says (x,y)
such that x
doesn't belongs to B
and y
doesn't belongs to C
, isn't that a contradiction or am I missing something here ?