1. What Is the Algebra of Sets?
Algebra of sets is a collection of rules and laws that describe how set operations behave. These rules make it easier to simplify expressions involving unions, intersections, differences, and complements.
The laws work just like algebraic rules for numbers, but here we apply them to sets.
2. Commutative Laws
The order of sets does not matter for union and intersection.
2.1. Laws
\( A \cup B = B \cup A \)
\( A \cap B = B \cap A \)
2.2. Example
If A = {1,2} and B = {2,3}, then A ∪ B = {1,2,3} and B ∪ A = {1,2,3}.
3. Associative Laws
The grouping of sets does not change the result.
3.1. Laws
\( (A \cup B) \cup C = A \cup (B \cup C) \)
\( (A \cap B) \cap C = A \cap (B \cap C) \)
3.2. Example
No matter how you group the sets, the union or intersection stays the same.
4. Distributive Laws
Union distributes over intersection and vice versa.
4.1. Laws
\( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \)
\( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
5. Identity Laws
The universal set U and the empty set ∅ act like identity elements for operations.
5.1. Laws
\( A \cup \emptyset = A \)
\( A \cap U = A \)
\( A \cup U = U \)
\( A \cap \emptyset = \emptyset \)
6. Idempotent Laws
Combining a set with itself does not change it.
6.1. Laws
\( A \cup A = A \)
\( A \cap A = A \)
7. Complement Laws
The complement of a set reverses membership.
7.1. Laws
\( A \cup A' = U \)
\( A \cap A' = \emptyset \)
\( (A')' = A \)
8. De Morgan’s Laws
These laws help simplify expressions involving complements of unions and intersections.
8.1. Laws
\( (A \cup B)' = A' \cap B' \)
\( (A \cap B)' = A' \cup B' \)
8.2. Example
The complement of a union becomes the intersection of complements, and vice versa.
9. Absorption Laws
These laws reduce combined expressions to simpler forms.
9.1. Laws
\( A \cup (A \cap B) = A \)
\( A \cap (A \cup B) = A \)
10. Difference and Complement Relations
Difference can be written using intersection and complement.
10.1. Laws
\( A - B = A \cap B' \)
11. Important Points
A few things to remember about set algebra:
11.1. Key Ideas
- Set algebra rules help simplify complex expressions.
- Union and intersection behave like addition and multiplication in many ways.
- De Morgan’s laws are especially important for complements.
- The universal set and empty set act like special identity elements.