Question of Venn Diagrams and Subsets on a Book

Question

In this book, Section 1.2.1 on page 7 introduce Sets and a diagram on page 8

Here, I am confused about the hightlighted area.

Is there an error here? Should

A ∩ Bc = {s : s ∈ A and s ∈ B}

be

A ∩ Bc = {s : s ∈ A and s ∈ Bc}

Isn't A ∩ Bc the waning cresent(moon phase term) of A?

Answer 1

You are correct. It should say

$$A\cap B^c = \{s : s \in A \text{ and } s \in B^c \}$$

or, equivalently,

$$A\cap B^c = \{s : s \in A \text{ and } s \notin B \}.$$

(Note that the equivalence of these definitions depends on the fact that $B^c$ is defined with respect to a larger set $S$ that contains $A$.)