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We are given sets \( A \), \( B \), \( C \), and \( T \) such that:

- \( T = \{1, 2, 3, 4, 5, 6\} \)
- \( A \Delta C = \{1, 2\} \)
- \( B \Delta C = \{5, 6\} \)
- \( A \cap C = B \cap C = \{3, 4\} \)

### Let's break this down:

1. \( A \Delta C \) (the symmetric difference of \( A \) and \( C \)) is the set of elements in either \( A \) or \( C \), but not in both. Hence, \( A \Delta C = \{1, 2\} \) implies:
   - \( 1, 2 \in A \) or \( 1, 2 \in C \), but not in both.

2. \( B \Delta C \) is similarly defined as the symmetric difference of \( B \) and \( C \). Given \( B \Delta C = \{5, 6\} \), we know:
   - \( 5, 6 \in B \) or \( 5, 6 \in C \), but not in both.

3. \( A \cap C = B \cap C = \{3, 4\} \) tells us that:
   - Both \( A \) and \( B \) share the common elements \( 3, 4 \) with \( C \).

### Steps to find \( A \), \( B \), and \( C \):

- \( A \Delta C = \{1, 2\} \) means that the elements \( 1, 2 \) are in either \( A \) or \( C \), but not both. Since \( A \cap C = \{3, 4\} \), we know that \( 1, 2 \) cannot be in \( C \), so they must be in \( A \).
  
- Similarly, \( B \Delta C = \{5, 6\} \) implies that \( 5, 6 \) are in either \( B \) or \( C \), but not both. Since \( B \cap C = \{3, 4\} \), \( 5, 6 \) cannot be in \( C \), so they must be in \( B \).

- \( A \cap C = \{3, 4\} \) and \( B \cap C = \{3, 4\} \) mean that \( C \) contains the elements \( 3 \) and \( 4 \) for sure.

### Conclusion:

From these conditions, we can now determine the sets:

- \( A = \{1, 2, 3, 4\} \)
- \( B = \{3, 4, 5, 6\} \)
- \( C = \{3, 4\} \)

Let me know if you'd like further clarifications, or if you have any questions!

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Here are 5 related questions to deepen understanding:

1. What is the symmetric difference \( A \Delta B \) for the determined sets?
2. How does the concept of set intersection help solve this problem?
3. What are the properties of the symmetric difference operator?
4. How would this problem change if we had \( A \cap B = \emptyset \)?
5. Can you describe the union \( A \cup B \) based on these sets?

**Tip:** When dealing with set operations like symmetric difference, try to visualize each operation using Venn diagrams. This can help clarify which elements belong to which sets!

Determine A, B, C ⊆ T and AΔB, if T = {1, 2, 3, 4, 5, 6}, AΔC = {1, 2}, BΔC = {5, 6}, A ∩ C = B ∩ C = {3, 4}.

Solve a set theory problem where sets A, B, and C are subsets of T = {1, 2, 3, 4, 5, 6}. Given conditions such as AΔC = {1, 2}, BΔC = {5, 6}, and A ∩ C = B ∩ C = {3, 4}, we find A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, and C = {3, 4}.

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