Circles A, B, and C overlap. x is in the overlap of circles A and B.

Question

Which of the following are true about x? Check all that apply.

x ∉ A
x ∈ B
x ∉ C
x ∈ A ⋃ B
x ∈ A ⋃ C
x ∈ A ⋂ B

Answer 1

To analyze the statements about the point \( x \) that is in the overlap of circles A and B, we can break down the statements one by one.

1. \( x \notin A \) - This statement is false because \( x \) is in the overlap of circles A and B, which means \( x \) must be in A.

2. \( x \in B \) - This statement is true because \( x \) is in the overlap of circles A and B, meaning \( x \) is inside circle B.

3. \( x \notin C \) - This statement can be either true or false depending on the position of circle C in relation to circles A and B. Since we do not have specific information about circle C's position relative to circles A and B, we cannot determine the truth of this statement.

4. \( x \in A \cup B \) - This statement is true because being in the overlap of circles A and B means \( x \) is in at least one of the circles A or B.

5. \( x \in A \cup C \) - This statement can be either true or false depending again on the position of circle C. Since we have no information about how circle C overlaps or relates to circles A and B, we cannot determine the truth of this statement.

6. \( x \in A \cap B \) - This statement is true since \( x \) is in the overlap of circles A and B, which is explicitly defined as the intersection \( A \cap B \).

Based on this analysis, the statements that are definitely true about \( x \) are:

• \( x \in B \)
• \( x \in A \cup B \)
• \( x \in A \cap B \)

The statements \( x \notin C \) and \( x \in A \cup C \) cannot be concluded without further information about circle C.