https://file.aiquickdraw.com/mathbot/file/17272540796590pafyl0a.jpg

The problem provided is in Indonesian and reads:

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**Tugas Terstruktur:**

*Misalkan A, B, C, dan D adalah himpunan. Buktikan bahwa \( A \cup (B \cap (C \cap D)) \) = ... Cari dualnya dan buat table Pembuktikan dengan menggunakan tabel keanggotaan.*

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**Translation:**

**Structured Task:**

*Assume that \( A \), \( B \), \( C \), and \( D \) are sets. Prove that \( A \cup (B \cap (C \cap D)) \) equals a certain expression, find its dual, and construct the proof using a membership table (truth table).*

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### Steps to solve the problem:

1. **Simplify the Expression**:
   You are tasked with proving that the union of set \( A \) with the intersection of sets \( B \), \( C \), and \( D \) simplifies to some expression.
   \[
   A \cup (B \cap (C \cap D))
   \]
   The goal is to break this expression down using set identities or simplifications.

2. **Find the Dual**:
   The dual of a set expression is found by switching unions (\( \cup \)) with intersections (\( \cap \)), and vice versa.

   Thus, the dual of \( A \cup (B \cap (C \cap D)) \) would be:
   \[
   A \cap (B \cup (C \cup D))
   \]

3. **Construct a Membership Table**:
   This is a truth table that checks all possible memberships of elements in the sets \( A \), \( B \), \( C \), and \( D \). For every possible combination of membership (element being in or out of the set), you compute whether the element is in the union and intersection expressions.

### To solve this, we will:

- Write out the truth table with all possible combinations of membership (True/False) for the sets.
- Verify both the original expression and its dual using the truth table.

Do you want me to walk through the entire solution and generate the truth table for this?

Misalkan A, B, C, dan D adalah himpunan. Buktikan bahwa A ∪ (B ∩ (C ∩ D)) = ... cari dualnya dan buat tabel Pembuktikan dengan menggunakan tabel keanggotaan.

This problem explores set theory, proving the union of sets and finding the dual. Detailed solutions using membership tables are provided, along with concepts like De Morgan's Law and distributive laws in set theory. Ideal for university students studying discrete mathematics.

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