1. What Are Standard Set Identities?
Standard set identities are a collection of well-known rules that describe how sets behave under operations like union, intersection, complement, and difference. These identities help simplify expressions and make it easier to solve problems involving sets.
They work for all sets within the same universal set and form the foundation of set algebra.
2. Idempotent Laws
Combining a set with itself does not change it.
2.1. Formulas
\( A \cup A = A \)
\( A \cap A = A \)
3. Commutative Laws
The order of sets does not change the result of union or intersection.
3.1. Formulas
\( A \cup B = B \cup A \)
\( A \cap B = B \cap A \)
4. Associative Laws
Grouping does not affect the result when using only union or only intersection.
4.1. Formulas
\( (A \cup B) \cup C = A \cup (B \cup C) \)
\( (A \cap B) \cap C = A \cap (B \cap C) \)
5. Distributive Laws
Union distributes over intersection and intersection distributes over union.
5.1. Formulas
\( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \)
\( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
6. Identity Laws
The universal set U and the empty set ∅ act as identities for certain operations.
6.1. Formulas
\( A \cup \emptyset = A \)
\( A \cap U = A \)
7. Domination Laws
Union with U or intersection with ∅ leads to fixed results.
7.1. Formulas
\( A \cup U = U \)
\( A \cap \emptyset = \emptyset \)
8. Complement Laws
Every element belongs either to a set or its complement.
8.1. Formulas
\( A \cup A' = U \)
\( A \cap A' = \emptyset \)
\( (A')' = A \)
9. Difference and Complement Relation
Set difference can be expressed using intersection and complement.
9.1. Formula
\( A - B = A \cap B' \)
10. Absorption Laws
These laws simplify complex expressions involving both union and intersection.
10.1. Formulas
\( A \cup (A \cap B) = A \)
\( A \cap (A \cup B) = A \)
11. De Morgan’s Laws
These laws connect complements with union and intersection.
11.1. Formulas
\( (A \cup B)' = A' \cap B' \)
\( (A \cap B)' = A' \cup B' \)
12. Important Points
Things to remember about standard set identities:
12.1. Key Ideas
- These identities help simplify expressions involving sets.
- They work for all sets under the same universal set.
- Many identities come in dual pairs.
- Useful in proofs, problem-solving, and logical reasoning.