 Let A and B be subsets of a universal set U. Show that A ⊆ B if and only if complement B
⊆ complement A.


To prove that \( A \subseteq B \) if and only if \( B^c \subseteq A^c \), we must demonstrate both directions of the equivalence:

1. **If \( A \subseteq B \), then \( B^c \subseteq A^c \).**
2. **If \( B^c \subseteq A^c \), then \( A \subseteq B \).**

Here is the detailed proof:

---

### 1. **If \( A \subseteq B \), then \( B^c \subseteq A^c \):**
- Suppose \( A \subseteq B \). This means that for every element \( x \in U \):
  \[
  x \in A \implies x \in B.
  \]
- The contrapositive of this implication is:
  \[
  x \notin B \implies x \notin A.
  \]
- In terms of complements, this means:
  \[
  x \in B^c \implies x \in A^c.
  \]
- Therefore, every element of \( B^c \) is also an element of \( A^c \), so:
  \[
  B^c \subseteq A^c.
  \]

---

### 2. **If \( B^c \subseteq A^c \), then \( A \subseteq B \):**
- Suppose \( B^c \subseteq A^c \). This means that for every element \( x \in U \):
  \[
  x \in B^c \implies x \in A^c.
  \]
- Equivalently:
  \[
  x \notin B \implies x \notin A.
  \]
- Taking the contrapositive:
  \[
  x \in A \implies x \in B.
  \]
- Therefore, every element of \( A \) is also an element of \( B \), so:
  \[
  A \subseteq B.
  \]

---

### Conclusion:
Since both directions have been proven, we conclude that:
\[
A \subseteq B \iff B^c \subseteq A^c.
\]

---

Would you like more examples or details? Here are some questions to deepen understanding:

1. How does this proof rely on the contrapositive of logical implications?
2. Can you provide a concrete example where \( A \subseteq B \) and verify that \( B^c \subseteq A^c \)?
3. How do these subset relationships work in the context of Venn diagrams?
4. What happens if \( A \) and \( B \) are disjoint sets?
5. How is the concept of complements used in Boolean algebra or set theory?

**Tip:** Always try visualizing set relationships using Venn diagrams to solidify understanding.

Let A and B be subsets of a universal set U. Show that A ⊆ B if and only if complement B ⊆ complement A.

Explore the proof that A ⊆ B if and only if complement B ⊆ complement A in set theory. This step-by-step guide covers subset relationships, set complements, and the role of contrapositive logic. Perfect for high school students in Grades 10-12 studying advanced set theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation