Question concerning Boolean Algebra of pairs

Question

I am reading a book "Boolean Algebra" by R.L. Goodstein.

In section 2.19 of the chapter "Self dual system of axioms", I am not able to comprehend what the author is trying to say in the first line of the very section.

It says, "If we select the subset of pairs $(A, A^c) $ from the set of pairs $(A, B) $ $\dots $".

What is meant by "the subset of pairs $(A, A^c) $ from the set of pairs $(A, B)$." ?

Here is the picture.

A detailed explanation that what is meant by the author here would be helpful.

Answer 1

Some pairs $(A,B)$ have the property that $B=A^c$; others don't. The set of pairs of the form $(A,A^c)$ forms a subset of the set of all pairs. Put another way, we have two sets of pairs in question:

• The set of all pairs.

• The set of pairs where the second coordinate is the complement of the first coordinate.

The point is that the latter is a subset of the former.