Question regarding Boolean Algebra of pairs

Question

Why is the union of $(A_1, B_1), (A_2, B_2)$ defined as $(A_1 \cup A_2, B_1 \cap B_2)$ and why is the intersection of $(A_1, B_1), (A_2, B_2)$ defined as $(A_1 \cap A_2, B_1 \cup B_2)$?

What is the reason or logic of considering the aforementioned and why don't we define the union of $(A_1, B_1), (A_2, B_2)$ as $(A_1 \cup A_2, B_1 \cup B_2)$ and intersection of $(A_1, B_1), (A_2, B_2)$ as $(A_1 \cap A_2, B_1 \cap B_2)$ ?

And why do we define the zero and unit pairs $(0, 1)$ and $(1, 0)$ respectively ?

A detailed explanation would be helpful.

Answer 1

There are good reasons for choosing these definitions.

Let $\mathscr{A}$ be the base algebra and $\mathscr{P}$ the algebra of pairs. Then you can consider the map $$ j\colon\mathscr{P}\to\mathscr{P},\qquad j\bigl((A,B)\bigr)=(B,A) $$ Now try to do \begin{align} j\bigl((A_1,B_1)\cup(A_2,B_2)\bigr) &=j\bigl((A_1\cup A_2,B_1\cap B_2)\bigr)\\ &=(B_1\cap B_2,A_1\cup A_2)\\ &=(B_1,A_1)\cap(B_2,A_2)\\ &=j\bigl((A_1,B_1)\bigr)\cap j\bigl((A_2,B_2)\bigr) \end{align} Similarly, $$ j\bigl((A_1,B_1)\cap(A_2,B_2)\bigr)=j\bigl((A_1,B_1)\bigr)\cup j\bigl((A_2,B_2)\bigr) $$ Therefore, $j$ is a bijection that exchanges $\cup$ with $\cap$, so the Boolean algebra $\mathscr{P}$ is self-dual.

Answer 2

In mathematics you can define things any which way you want. The question is how these definitions are going to be used, i.e. what practical purpose they serve. So ... keep on reading and see how the author uses these definitions. But we can't tell you what is going on in the author's mind when coming up with those definitions.

Now, given the basic definition of $\cap$ and $\cap$, it does make sense to consider $(0,1)$ to be the zero element (i.e. identity element for $\cup$), and $(1,0)$ to be the unit element (i.e. identity element for $\cap$) since for any $A$ and $B$:

$(A,B) \cup (0,1) = (A \cup 0, B \cap 1) = (A,B)$

and

$(A,B) \cap (1,0) = (A \cap 1, B \cup 0) = (A,B)$