Question

shade the portion of the diagram that illustrates the set. (a∪b)∩ĉ use the graphing tool to graph the set. click to enlarge graph

Explanation:

Step1: Analyze \( A \cup B \)

First, find the union of sets \( A \) and \( B \). The union \( A \cup B \) includes all regions inside circle \( A \), inside circle \( B \), or in their intersection. So this is the combined area of circles \( A \) and \( B \) (including their overlap).

Step2: Analyze \( \overline{C} \)

The complement of set \( C \), \( \overline{C} \), is all regions in the universal set \( U \) that are not inside circle \( C \).

Step3: Find \( (A \cup B) \cap \overline{C} \)

The intersection of \( A \cup B \) and \( \overline{C} \) is the region that is in \( A \cup B \) (from Step 1) and also in \( \overline{C} \) (from Step 2). So we shade the parts of circles \( A \) and \( B \) (including their overlap) that do not overlap with circle \( C \). In the Venn diagram, this means shading the areas of \( A \) (not overlapping with \( C \)), \( B \) (not overlapping with \( C \)), and the overlap of \( A \) and \( B \) (not overlapping with \( C \)).

Answer:

To shade \( (A \cup B) \cap \overline{C} \), shade the regions:

• Inside circle \( A \) but not inside circle \( C \),
• Inside circle \( B \) but not inside circle \( C \),
• Inside the intersection of \( A \) and \( B \) but not inside circle \( C \).