pizza or burger or both. Ten students answered that they don't like pizza nor burgers at all. Twenty students like both pliza and burgers and 12 stu- dents only liked pizza. A. Complete the Venn diagram. Pizza Burger

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UpStudy AI · Accepted Answer

To solve this problem, we need to analyze the information given about the students' preferences for pizza and burgers. Let's summarize the known conditions:

1. **Students who don't like pizza nor burgers:** 10 students
2. **Students who like both pizza and burgers:** 20 students
3. **Students who only like pizza:** 12 students

Let's denote:
- $$ P $$: the set of students who like pizza
- $$ B $$: the set of students who like burgers
- $$ n(P) $$: the number of students who like pizza
- $$ n(B) $$: the number of students who like burgers
- $$ n(P \cap B) $$: the number of students who like both pizza and burgers
- $$ n(P \setminus B) $$: the number of students who only like pizza
- $$ n(B \setminus P) $$: the number of students who only like burgers
- $$ n(P \cup B) $$: the total number of students who like either pizza or burgers

### Step 1: Calculate the total number of students who like pizza or burgers

From the information provided:
- Students who like only pizza: $$ n(P \setminus B) = 12 $$
- Students who like both: $$ n(P \cap B) = 20 $$

Thus, the total number of students who like pizza is:
$$
n(P) = n(P \setminus B) + n(P \cap B) = 12 + 20 = 32
$$

### Step 2: Determine the number of students who only like burgers

Let $$ n(B \setminus P) $$ be the number of students who only like burgers. We can express the total number of students who like burgers as:
$$
n(B) = n(B \setminus P) + n(P \cap B)
$$

### Step 3: Calculate the total number of students

Let $$ x $$ be the total number of students. We know:
$$
x = n(P \cup B) + \text{students who don't like either}
$$
$$
x = n(P \setminus B) + n(B \setminus P) + n(P \cap B) + 10
$$

Substituting the known values:
$$
x = 12 + n(B \setminus P) + 20 + 10
$$
$$
x = n(B \setminus P) + 42
$$

### Step 4: Find the number of students who only like burgers

To find $$ n(B \setminus P) $$, we need to express $$ n(B) $$ in terms of $$ x $$. We can rearrange the equation:
$$
n(B) = n(B \setminus P) + 20
$$

Now, we can express $$ n(B \setminus P) $$ in terms of $$ x $$:
$$
n(B \setminus P) = x - 42
$$

### Step 5: Solve for $$ x $$

Since we have two equations involving $$ n(B \setminus P) $$:
1. $$ n(B \setminus P) = x - 42 $$
2. $$ n(B) = n(B \setminus P) + 20 $$

Substituting $$ n(B \setminus P) $$ into the second equation:
$$
n(B) = (x - 42) + 20 = x - 22
$$

### Step 6: Total students who like either pizza or burgers

Now we can express $$ n(P \cup B) $$:
$$
n(P \cup B) = n(P \setminus B) + n(B \setminus P) + n(P \cap B) = 12 + (x - 42) + 20
$$
$$
n(P \cup B) = x - 10
$$

### Step 7: Set up the equation

Now we can set up the equation:
$$
x = (x - 10) + 10
$$
This simplifies to:
$$
x = x
$$
This means we need to find a specific value for $$ n(B \setminus P) $$.

### Step 8: Finalize the Venn Diagram

We can summarize the findings:
- Students who like only pizza: 12
- Students who like both: 20
- Students who don't like either: 10
- Students who like only burgers: $$ n(B \setminus P) = x - 42 $$

Since we don't have a specific total number of students, we can represent the Venn diagram as follows:

- **Only Pizza:** 12
- **Both Pizza and Burgers:** 20
- **Only Burgers:** $$ n(B \setminus P) $$ (unknown without total students)
- **Neither:** 10

If you have a specific total number of students, we can calculate $$ n(B \setminus P) $$ directly.
To complete the Venn diagram: - **Only Pizza:** 12 - **Both Pizza and Burgers:** 20 - **Only Burgers:** Unknown (depends on total number of students) - **Neither:** 10

pizza or burger or both. Ten students answered that they don't like pizza nor burgers at all. Twenty students like both pliza and burgers and 12 stu- dents only liked pizza. A. Complete the Venn diagram. Pizza Burger

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