Let's break down and solve each part of the problem presented in the image:

Part (i)

We are given the following information:

• $|A| = 72$ (the cardinality of set $A$),
• $|B| = 28$ (the cardinality of set $B$),
• $|A \cap B| = 13$ (the cardinality of the intersection of sets $A$ and $B$).

We are asked to find the cardinality of $A \cup B$ (the union of sets $A$ and $B$).

The formula for the cardinality of the union of two sets is: $|A \cup B| = |A| + |B| - |A \cap B|$

Substituting the values we have: $|A \cup B| = 72 + 28 - 13 = 87$

Thus, the cardinality of $A \cup B$ is 87.

Part (ii)

We are given:

• $|A - B| = 45$ (the cardinality of the difference between sets $A$ and $B$),
• $|A \cup B| = 110$,
• $|A \cap B| = 15$.

We are asked to find $|B|$, the cardinality of set $B$.

We know the formula for the cardinality of the difference between two sets: $|A - B| = |A| - |A \cap B|$ Thus, solving for $|A|$: $|A| = |A - B| + |A \cap B| = 45 + 15 = 60$

Now, using the formula for the union of two sets: $|A \cup B| = |A| + |B| - |A \cap B|$ Substituting the known values: $110 = 60 + |B| - 15$ Simplifying: $110 = 45 + |B|$ $|B| = 110 - 45 = 65$

Thus, the cardinality of set $B$ is 65.

Part (iii)

We are given:

• $|A| = 33$,
• $|B| = 36$,
• $|C| = 28$,
• We are asked to find $|A \cup B \cup C|$.

The general formula for the union of three sets is: $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C|$ However, from the image and the problem statement, it seems like we're dealing with the Venn diagram to help us find the correct values for the intersections of the sets. Specifically, the diagram shows the overlapping areas of the three sets.

From the diagram, we can use the following intersections:

• $|A \cap B| = 5$,
• $|B \cap C| = 10$,
• $|A \cap C| = 15$,
• $|A \cap B \cap C| = 5$.

Now, applying the formula: $|A \cup B \cup C| = 33 + 36 + 28 - 5 - 10 - 15 + 5$ Simplifying: $|A \cup B \cup C| = 97 - 30 + 5 = 72$

Thus, the cardinality of $A \cup B \cup C$ is 72.

Do you need further explanation or details? Here are some related questions:

1. How is the cardinality of intersections of more than two sets derived?
2. What are the general properties of set operations like union and intersection?
3. How can a Venn diagram visually assist in solving problems involving three sets?
4. Can we apply similar methods to solve set operations with more than three sets?
5. How do we interpret $A - B$ in the context of a Venn diagram?

Tip: Always remember to apply the principle of inclusion and exclusion when calculating the union of multiple sets.