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### Problem A:
- There are 25 employees in total.
- 20 of them like iced tea.
- 15 of them like orange juice.

We are tasked to find the following:

#### a. How many like both iced tea and orange juice?

Using the principle of set theory:
Let:
- \( |A| = 20 \) (those who like iced tea),
- \( |B| = 15 \) (those who like orange juice),
- \( |A \cup B| = 25 \) (total employees).

We use the inclusion-exclusion formula:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
Substitute the known values:
\[
25 = 20 + 15 - |A \cap B|
\]
\[
25 = 35 - |A \cap B|
\]
\[
|A \cap B| = 10
\]

Thus, **10 people** like both iced tea and orange juice.

#### b. How many like iced tea only?

To find this, subtract those who like both from those who like iced tea:
\[
|A \text{ only}| = |A| - |A \cap B| = 20 - 10 = 10
\]

Thus, **10 people** like only iced tea.

#### c. How many like orange juice only?

Similarly, subtract those who like both from those who like orange juice:
\[
|B \text{ only}| = |B| - |A \cap B| = 15 - 10 = 5
\]

Thus, **5 people** like only orange juice.

---

### Problem B:
This is a Venn diagram problem involving the set of counting numbers to 20. Let's answer the questions based on the Venn diagram:

#### 1. What are the members of the universal set?
The universal set \( U \) contains the counting numbers from 1 to 20:
\[
U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}
\]

#### 2. What are the members of set A?
From the Venn diagram, set A contains:
\[
A = \{1, 2, 3, 4, 6, 12\}
\]

#### 3. What are the members of set B?
From the diagram, set B contains:
\[
B = \{3, 6, 9, 12, 18\}
\]

#### 4. List the members that intersect.
The intersection of set A and set B (those common to both sets) is:
\[
A \cap B = \{3, 6, 12\}
\]

#### 5. What members are in the universal set but not a member of set A nor set B?
These are the elements of \( U \) that are neither in set A nor set B. From the diagram, these members are:
\[
U - (A \cup B) = \{5, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 20\}
\]

---

Would you like more details on any part of the solution, or do you have any specific questions?

### 5 Related Questions:
1. How do you calculate the union of two sets in set theory?
2. Can you explain the inclusion-exclusion principle in more detail?
3. What happens when two sets have no common elements (disjoint sets)?
4. How would the solution change if the total number of employees was different?
5. How do you interpret Venn diagrams in real-world problems?

### Tip:
Always use the inclusion-exclusion principle when dealing with overlapping sets to avoid double-counting elements.

Illustrate the Venn Diagram and solve the problem. There are 25 employees in a company. They like either iced tea or orange juice. If 20 of them like iced tea and 15 of them like orange juice: a. How many like both? b. How many like iced tea only? c. How many like orange juice only?

Solve a set theory problem involving 25 employees where 20 like iced tea, 15 like orange juice, and some like both. Learn how to use the inclusion-exclusion principle to find how many like each drink.

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