Question

Draw Venn diagrams for \(A \cap(B \cup C)\) and \((A \cap B) \cup(A \cap C) .\) Based on your drawings, do you think \(A \cap(B \cup C)=(A \cap B) \cup(A \cap C) ?\)

Short answer

Whether \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \) can only be confirmed by accurately drawing and comparing the two Venn diagrams.

Step-by-step solution

Draw the Venn Diagams

Before starting, remember that a Venn diagram for three sets A, B, and C, usually involves three overlapping circles. The overlap regions represent the intersection of the sets. Begin with constructing two Venn diagrams. The first diagram will represent \( A \cap (B \cup C) \). The second diagram will represent \( (A \cap B) \cup (A \cap C) \). In diagram one mark the area for A intersected with the union of B and C, which essentially is the overlap region of A with both B and C. In the second diagram, mark the area for A intersected with B and A intersected with C, this corresponds to the overlap of A and B, and similarly, A and C.

Compare the Diagrams

Inspect both diagrams visually. The marked regions in both diagrams must look exactly the same if the two expressions are equivalent.

Conclusion

Based on the visual similarity of the Venn diagrams drawn in step one, if the diagrams of \( A \cap (B \cup C) \) and \( (A \cap B) \cup (A \cap C) \) are identical, then it is concluded that they are equivalent; otherwise, they are not.