To solve the given Venn diagram question, we need to shade the region corresponding to $A \cup B'$.

• $A \cup B'$ means the union of set $A$ with the complement of set $B$.
• The complement of $B$ ($B'$) refers to everything outside of set $B$ in the universal set $\mathcal{E}$.

Step-by-Step Process:

1. Set $B'$ includes all elements that are not in set $B$. This means everything outside of set $B$, including the area inside set $A$ but outside $B$.
2. Set $A \cup B'$ includes everything that is either in set $A$ or in $B'$. This will include:
   • All of set $A$.
   • Everything outside of set $B$ (including the part of $A$ outside $B$, and areas in the universal set outside of both sets).

Shading:

• Shade all of set $A$ because $A \cup B'$ includes all of set $A$.
• Also shade the areas outside of set $B$, including the part that lies outside both sets.

Thus, the region $A \cup B'$ will include:

• The entirety of set $A$,
• Everything outside of set $B$ (except the part inside $B$).

Would you like to go through more detailed examples or have further clarifications?

5 Relative Questions:

1. What is the difference between $A \cap B$ and $A \cup B$?
2. How would you represent $B \cup A'$ in a Venn diagram?
3. What does the complement of a set in a Venn diagram represent?
4. Can a Venn diagram have more than two sets? How would you represent three sets?
5. How would the region $A' \cap B$ be shaded on a Venn diagram?

Tip:

When solving Venn diagram problems, it's helpful to think of each set and its complement as separate regions, making it easier to determine which areas need to be shaded based on the union or intersection.