Let’s analyze and represent each of these set theory expressions using Venn diagrams.

1. (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C)

Left-Hand Side (LHS): (A ∩ B) ∪ C

• A ∩ B: The intersection of sets A and B, i.e., elements common to both.
• (A ∩ B) ∪ C: The union of (A ∩ B) with C, which means all elements that are in either A ∩ B or C.

Right-Hand Side (RHS): (A ∪ C) ∩ (B ∪ C)

• A ∪ C: The union of A and C, i.e., all elements in A or C.
• B ∪ C: The union of B and C, i.e., all elements in B or C.
• (A ∪ C) ∩ (B ∪ C): The intersection of (A ∪ C) and (B ∪ C), i.e., elements that are common in both (A ∪ C) and (B ∪ C).

Both LHS and RHS represent the same region in the Venn diagram, as the union and intersection behave symmetrically in this case.

2. (A ∩ B)c = (Ac ∪ Bc)

Left-Hand Side (LHS): (A ∩ B)c

• A ∩ B: The intersection of A and B, i.e., elements common to both.
• (A ∩ B)c: The complement of A ∩ B, which means all elements outside the intersection of A and B.

Right-Hand Side (RHS): (Ac ∪ Bc)

• Ac: The complement of A, i.e., all elements outside A.
• Bc: The complement of B, i.e., all elements outside B.
• Ac ∪ Bc: The union of Ac and Bc, i.e., all elements that are either outside A or outside B.

Both LHS and RHS represent the same region, as the complement of the intersection of A and B is the same as the union of the complements of A and B.

3. If A ⊂ B, then Bc ⊂ Ac

Hypothesis: A ⊂ B

• If A is a subset of B, this means all elements of A are also in B.

Conclusion: Bc ⊂ Ac

• Bc: The complement of B, i.e., all elements outside B.
• Ac: The complement of A, i.e., all elements outside A.

Since A ⊂ B, all elements outside B (Bc) must also be outside A (Ac), so Bc ⊂ Ac.

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Would you like detailed Venn diagrams for these relationships? Let me know!

Here are 5 questions related to the concepts:

1. What is the difference between union and intersection of sets?
2. How does De Morgan's law relate to complements in set theory?
3. Can you find a counterexample where $A ⊈ B$ and $B^c ⊈ A^c$?
4. How would you use Venn diagrams to prove distributive properties of sets?
5. How do complements affect the relationships between subsets?

Tip: When working with set theory, always check the basic definitions of union, intersection, and complement, as many logical manipulations rely on these fundamentals.