1. What Are the Distributive Laws?
The distributive laws explain how union and intersection interact with each other. They allow one operation to be distributed over the other, similar to how multiplication distributes over addition in arithmetic.
These laws help simplify complex set expressions.
2. Union Distributes Over Intersection
The union of a set with the intersection of two sets is the same as taking the union individually and then intersecting the results.
2.1. Formula
\( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \)
2.2. Example
If A = {1}, B = {1,2}, C = {1,3}, then:
A \cup (B \cap C) = \{1\}
(A \cup B) \cap (A \cup C) = \{1\}
3. Intersection Distributes Over Union
The intersection of a set with the union of two sets works in a similar way.
3.1. Formula
\( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
3.2. Example
If A = {2,3}, B = {3,4}, C = {3,5}, then:
A \cap (B \cup C) = \{3\}
(A \cap B) \cup (A \cap C) = \{3\}
4. Why Distributive Laws Matter
These laws are essential when simplifying expressions involving both union and intersection. They help rewrite expressions into clearer or more useful forms.
5. Important Points
Things to keep in mind about distributive laws:
5.1. Key Ideas
- Union can distribute over intersection.
- Intersection can distribute over union.
- These laws make complex set expressions easier to manage.
- They work for all sets A, B, and C.