5 Questions
¿Cuál es el complemento del conjunto universal U?
El conjunto vacío ∅
¿Cuál es la propiedad del complemento de un conjunto A?
(A')' = A
¿Cuál es la relación entre el complemento de la unión de dos conjuntos y el complemento de la intersección de los mismos conjuntos?
(A ∪ B)' = A' ∩ B'
¿Cuál es el resultado de encontrar el complemento del conjunto A = {1, 2, 3} en el conjunto universal U = {1, 2, 3, 4, 5, 6}?
{4, 5, 6}
¿Cuál es la relación entre el complemento de la intersección de dos conjuntos y el complemento de la unión de los mismos conjuntos?
(A ∩ B)' = A' ∪ B'
Study Notes
Complement of a Set
Definition
The complement of a set A, denoted as A' or Ac, is the set of all elements in the universal set U that are not in A.
Properties
- The complement of the universal set U is the empty set ∅.
- The complement of the empty set ∅ is the universal set U.
- For any set A, (A')' = A (double complement law).
- For any sets A and B, (A ∪ B)' = A' ∩ B'.
Operations
- De Morgan's laws:
- (A ∪ B)' = A' ∩ B'
- (A ∩ B)' = A' ∪ B'
- The complement of the union of two sets is the intersection of their complements.
- The complement of the intersection of two sets is the union of their complements.
Examples
- If the universal set U = {1, 2, 3, 4, 5, 6} and A = {1, 2, 3}, then A' = {4, 5, 6}.
- If the universal set U = {a, b, c, d, e} and B = {a, c, e}, then B' = {b, d}.
Learn about the definition, properties, and operations of the complement of a set in set theory, including De Morgan's laws and examples.
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