In a survey of 100 students, 78 liked math, 62 liked science, and 18 liked math but not science.
How many students do not like either subject

Let's solve this problem step by step using set theory.

### Given:
- Total number of students surveyed, \( n = 100 \)
- Number of students who liked math, \( |M| = 78 \)
- Number of students who liked science, \( |S| = 62 \)
- Number of students who liked math but not science, \( |M \setminus S| = 18 \)

### Objective:
We need to find the number of students who do not like either subject.

### Step 1: Find the number of students who like both math and science.
The students who liked both math and science can be found using the following formula:
\[
|M \cap S| = |M| - |M \setminus S|
\]
Substitute the values:
\[
|M \cap S| = 78 - 18 = 60
\]

### Step 2: Find the number of students who like either math or science or both.
The number of students who liked either math or science or both can be found using the principle of inclusion-exclusion:
\[
|M \cup S| = |M| + |S| - |M \cap S|
\]
Substitute the values:
\[
|M \cup S| = 78 + 62 - 60 = 80
\]

### Step 3: Find the number of students who do not like either subject.
The number of students who do not like either math or science is:
\[
n - |M \cup S| = 100 - 80 = 20
\]

### Final Answer:
20 students do not like either subject.

Would you like more details or have any questions?

Here are some related questions:
1. What is the principle of inclusion-exclusion?
2. How would the answer change if the number of students who liked both subjects were given directly?
3. What if some students were surveyed but didn't answer?
4. How can Venn diagrams help in solving these kinds of problems?
5. Can we determine how many students liked only science?

**Tip:** In problems involving sets and overlapping groups, using a Venn diagram can be a helpful visual aid to organize information and solve the problem systematically.

Explore the solution to a problem involving students liking math and science, with step-by-step explanations using set theory and the principle of inclusion-exclusion. Learn how to determine the number of students who do not like either subject. Suitable for students in Grades 9-12.

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