Math Problem Statement

Indicate where the given set \( A \cap B \cap C^c \) is located on the Venn diagram provided.

Solution

The problem requires finding the region that represents ABCcA \cap B \cap C^c, where:

  • ABA \cap B is the intersection of sets AA and BB (the area common to both AA and BB).
  • CcC^c represents the complement of CC, meaning the area outside set CC.

To solve this:

  1. First, locate the region where sets AA and BB overlap. This is the intersection area between the two circles AA and BB.
  2. Next, find the part of this intersection that is outside set CC. This excludes any part of ABA \cap B that also intersects with CC.

In the Venn diagram provided, the correct region to select will be the portion of the ABA \cap B intersection that lies outside CC.

From the image, this corresponds to the section in the middle that overlaps between AA and BB but does not intersect with CC. This should be only one region on the diagram.

Would you like more detailed steps or have specific questions?


Here are some related questions:

  1. How would the shaded region change if the set were ABCA \cap B \cap C instead?
  2. What area would represent ABcCA \cap B^c \cap C in the same diagram?
  3. How is CcC^c different in terms of shading compared to CC?
  4. How would you represent the union ABCcA \cup B \cup C^c in a Venn diagram?
  5. What would the region look like for (AC)(BCc)(A \cap C) \cup (B \cap C^c)?

Tip: Always identify each region individually and then combine based on the intersection or union required by the problem.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Intersection
Complement

Formulas

\( A \cap B \cap C^c \) represents the intersection of A and B and the complement of C

Theorems

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Suitable Grade Level

Grades 9-12