1. What Are De Morgan’s Laws?
De Morgan’s laws describe how complements interact with unions and intersections. They provide a way to break down the complement of a combined set into simpler parts.
These laws are used widely in set theory, logic, and probability.
2. First De Morgan’s Law
The complement of the union of two sets is the same as the intersection of their complements.
2.1. Formula
\( (A \cup B)' = A' \cap B' \)
2.2. Meaning
Everything outside both A and B is the same as what is outside A and also outside B.
2.3. Example
If U = {1,2,3,4,5}, A = {1,2}, B = {2,3}, then:
(A \cup B)' = \{4,5\}
A' \cap B' = \{4,5\}
3. Second De Morgan’s Law
The complement of the intersection of two sets is the same as the union of their complements.
3.1. Formula
\( (A \cap B)' = A' \cup B' \)
3.2. Meaning
Anything not in both A and B is the same as anything outside A or outside B.
3.3. Example
If U = {1,2,3,4,5}, A = {1,2}, B = {2,3}, then:
(A \cap B)' = \{1,3,4,5\}
A' \cup B' = \{1,3,4,5\}
4. Why De Morgan’s Laws Matter
These laws are powerful tools for simplifying set expressions. They turn complicated complements into easier union or intersection expressions, making problem-solving smoother.
5. Important Points
Key ideas to remember:
5.1. Key Ideas
- The complement of a union becomes the intersection of complements.
- The complement of an intersection becomes the union of complements.
- Useful in proofs, logic, and probability problems.
- They work for any sets A and B within the same universal set.