1. What Is the Dual of Set Statements?
The dual of a set statement is another statement formed by interchanging union and intersection and replacing the universal set with the empty set, and vice versa. The idea of duality shows that many set identities come in pairs, each being a mirror image of the other.
This principle helps identify relationships and quickly create new valid statements from known ones.
2. How Duality Works
To form the dual of any valid set statement, follow these two rules:
2.1. Rule 1: Swap Operations
- Replace \( \cup \) with \( \cap \)
- Replace \( \cap \) with \( \cup \)
2.2. Rule 2: Swap Special Sets
- Replace the universal set \( U \) with the empty set \( \emptyset \)
- Replace \( \emptyset \) with \( U \)
3. Examples of Dual Pairs
Here are some common statements and their duals:
3.1. Example 1: Idempotent Law
Original: \( A \cup A = A \)
Dual: \( A \cap A = A \)
3.2. Example 2: Identity Law
Original: \( A \cup \emptyset = A \)
Dual: \( A \cap U = A \)
3.3. Example 3: Complement Law
Original: \( A \cup A' = U \)
Dual: \( A \cap A' = \emptyset \)
3.4. Example 4: Domination Law
Original: \( A \cup U = U \)
Dual: \( A \cap \emptyset = \emptyset \)
4. Why Duality Is Useful
Duality helps shorten work by allowing one identity to produce another instantly. Once you know a set law, its dual is automatically true. This is especially helpful in proofs and simplifying expressions.
5. How to Form Duals Yourself
To write the dual of any new expression:
5.1. Steps
- Rewrite the statement.
- Swap \( \cup \) ↔ \( \cap \).
- Swap \( U \) ↔ \( \emptyset \).
- Leave complements (′) and sets (A, B, C) unchanged.
5.2. Quick Example
Given:
\( (A \cup B)' = A' \cap B' \)
Dual:
\( (A \cap B)' = A' \cup B' \)
6. Important Points
Some key things to remember:
6.1. Key Ideas
- Duality pairs every set identity with another valid identity.
- The operations ∪ and ∩ are always swapped.
- The special sets U and ∅ also swap roles.
- Complements remain unchanged when finding duals.