Set theory

Question

i am not able to understand the proof.

How we reached statement 2 from statement 1 as I have marked in the picture (right side)

Can some one elaborate?

Answer 1

Let $a1 \in A$ and $a1  \notin B$, so then,

$(a1,a1)\notin (A \times B)\cup(B \times A)$, but $(a1,a1) \in (A \cup B) \times (A \cup B)$.

So we can clearly say that (i) is false. and you have caught the error in the proof correctly.👍