Shade the Venn diagram for the set B’ n (AUC)
The Correct Answer and Explanation is:
To shade the correct region for the set B′ ∩ (A ∪ C) on a Venn diagram, follow this description:
- B′ (read as “B complement”) refers to everything outside set B.
- A ∪ C (read as “A union C”) includes all elements that are in set A, in set C, or in both.
- The intersection B′ ∩ (A ∪ C) includes only the elements that are in A or C but not in B.
Step-by-step guide to shading:
- Start with three overlapping circles labeled A, B, and C inside a rectangle representing the universal set.
- Identify A ∪ C: Shade everything that belongs to circle A, circle C, or both. At this stage, do not worry about circle B.
- Identify B′: This is all the regions outside circle B.
- Now take the intersection: Find the area that lies both within A ∪ C and outside of B.
The correct shaded area is the portion of A and C that does not overlap with B. This includes:
- The part of A that is not shared with B.
- The part of C that is not shared with B.
- The area shared by A and C as long as it is not shared with B.
Explanation
In set theory, we often use Venn diagrams to visualize how different sets interact. The expression B′ ∩ (A ∪ C) asks us to find the intersection between the complement of B and the union of A and C. The union A ∪ C means we are interested in all areas that fall within either A or C, regardless of whether elements belong to both. This step expands our region of interest across two sets.
On the other hand, B′ requires us to exclude anything in set B. The complement of B includes everything in the universal set that is not part of B. Therefore, when we intersect this with A ∪ C, we are specifically looking for elements that fall within A or C, but strictly outside of B.
This kind of logical layering—taking a union first and then intersecting it with a complement—forces us to be precise with boundaries. The overlap between A and C may or may not be included, depending on its relationship with B. Only the regions that lie entirely outside of B while still being part of A or C will be shaded.
Such expressions are useful in probability, logic, and computer science when filtering data, formulating search queries, or setting logical access permissions. By visualizing B′ ∩ (A ∪ C), you practice not only basic set operations but also the higher-level thinking needed to combine them meaningfully.
