Let the Universal Set, S, have 143 elements. A and B are subsets of S. Set A contains 96 elements and Set B contains 20 elements. If Sets A and B have 3 elements in common, how many elements are in B but not in A?

We are asked to find how many elements are in Set \( B \) but not in Set \( A \). The relevant data is:

- The universal set \( S \) has 143 elements.
- Set \( A \) contains 96 elements.
- Set \( B \) contains 20 elements.
- Sets \( A \) and \( B \) have 3 elements in common.

### Steps to solve:

1. The number of elements in Set \( B \) but not in Set \( A \) is represented by:
   \[
   |B - A| = |B| - |A \cap B|
   \]
   where \( |B| \) is the number of elements in Set \( B \), and \( |A \cap B| \) is the number of elements in both \( A \) and \( B \) (the intersection of \( A \) and \( B \)).

2. From the problem, we know:
   - \( |B| = 20 \)
   - \( |A \cap B| = 3 \)

3. Plug these values into the formula:
   \[
   |B - A| = 20 - 3 = 17
   \]

Thus, there are **17 elements** in Set \( B \) but not in Set \( A \).

Would you like more details on how sets interact, or any other clarification?

Here are 5 related questions for further practice:
1. How many elements are in Set \( A \) but not in Set \( B \)?
2. What is the number of elements in \( A \cup B \) (the union of \( A \) and \( B \))?
3. If \( S \) represents the universal set, how many elements are in \( S - (A \cup B) \)?
4. If Set \( A \) contains 10 additional elements, how many elements will be in \( A - B \)?
5. How many elements are in \( S - B \), the complement of Set \( B \)?

**Tip**: The principle of set subtraction helps to determine elements unique to one set and is closely tied to the intersection of sets.

This problem explores how to find the number of elements in Set B but not in Set A, given a universal set with 143 elements, where Set A has 96 elements, Set B has 20 elements, and they share 3 elements in common. The solution involves applying basic set theory concepts, including set difference and intersection.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation