Use Venn diagrams to show
1. (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C)
2. (A ∩ B)c = (Ac ∪ Bc)
3. if A ⊂ B then Bc ⊂ Ac
where A, B and C are events defined over a sample space S.

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.

---

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.

Use Venn diagrams to show
1. (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C)
2. (A ∩ B)c = (Ac ∪ Bc)
3. if A ⊂ B then Bc ⊂ Ac where A, B and C are events defined over a sample space S.

Explore how to use Venn diagrams to prove key set theory relationships such as (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C), (A ∩ B)c = (Ac ∪ Bc), and A ⊂ B implies Bc ⊂ Ac. This problem involves concepts like union, intersection, subsets, and complements, and is ideal for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation