https://file.aiquickdraw.com/mathbot/file/1730133640645s7cr3pwq.jpg

To solve for \( X \) in the Venn diagram, let's analyze the given information and set up an equation based on the sets.

### Information Given:
- The universal set \( n(\xi) = 30 \).
- There are two sets, \( A \) and \( B \), with values within the sets and their intersections shown.
  - \( n(A) \cap n(B) = 4 \) (the intersection of \( A \) and \( B \)).
  - Elements only in \( A \): \( X \).
  - Elements only in \( B \): \( 7 \).
  - Elements outside both \( A \) and \( B \): \( 6 \).

### Setting Up the Equation:
The total number of elements in the universal set \( \xi \) is the sum of all distinct elements in and out of the sets. Thus:

\[
n(\xi) = X + 4 + 7 + 6
\]

Substitute \( n(\xi) = 30 \):

\[
30 = X + 4 + 7 + 6
\]

Simplify by combining constants:

\[
30 = X + 17
\]

Now, isolate \( X \):

\[
X = 30 - 17
\]
\[
X = 13
\]

### Solution:
The value of \( X \) is **13**.

Would you like further details on the solution or have any questions?

### Related Questions:
1. How do you determine the total of unique elements in overlapping sets?
2. What is the purpose of using Venn diagrams in set theory?
3. How does the universal set affect the calculation of unknowns within subsets?
4. How do we calculate intersections and unions in Venn diagrams?
5. How can this problem be extended to three overlapping sets?

### Tip:
When dealing with Venn diagrams, always start by analyzing each region separately to ensure all elements are accounted for accurately.

Study the diagram below and use it to find the value of X.

Learn how to find the unknown value X in a Venn diagram using set theory principles. Given n(ξ) = 30 and known values in the sets A and B, solve for X through algebraic techniques. This problem is ideal for students in Grades 7-9 studying set theory and Venn diagrams.

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