Let's solve the problem step by step.

Given:

• Universal Set ($U$): $U = \{5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}$

• Set $A$: $A = \{1, 2, 7, 8, 13, 14, 19, 20\}$ (Note: these elements are outside the universal set $U$).

• Set $B$: $B = \{6, 7, 8, 12, 13, 18, 19\}$

• Set $C$: $C = \{5, 11, 12, 15, 16, 18, 19\}$

Questions:

1. Number of elements in the set $(A \cup B \cup C)$

2. Number of elements in the set $(A \cap B \cap C)$

Step 1: Correcting Set $A$

Since $A$ contains elements not in the universal set $U$, we should consider only those elements in $A$ that are within $U$.

So, the corrected set $A$ (intersection of $A$ with $U$) is: $A = \{7, 8, 13, 14, 19, 20\}$

Step 2: Find $A \cup B \cup C$

• $A \cup B \cup C$ combines all elements from $A$, $B$, and $C$: $A \cup B \cup C = \{5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 18, 19, 20\}$

There are 13 elements in the union of sets $A$, $B$, and $C$.

Step 3: Find $A \cap B \cap C$

• $A \cap B \cap C$ contains only those elements that are in all three sets: $A \cap B \cap C = \{19\}$

There is 1 element in the intersection of sets $A$, $B$, and $C$.

Final Answers:

1. Number of elements in $A \cup B \cup C$: 13
2. Number of elements in $A \cap B \cap C$: 1

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Would you like further details or explanations? Here are some related questions:

1. What is the difference between the union and intersection of sets?
2. Can you find the number of elements in $A \cup B$ without using set $C$?
3. How would the problem change if sets $A$ and $B$ were subsets of the universal set $U$?
4. What are some practical applications of union and intersection in set theory?
5. How do Venn diagrams help in solving such set problems?

Tip: When working with sets, always ensure that the elements you're dealing with belong to the universal set, especially in problems involving unions and intersections.