1. De Morgan's Laws
De Morgan's Laws are fundamental rules in set theory that relate the complement of the union and intersection of sets. These laws are particularly useful in simplifying expressions involving sets.
De Morgan's First Law
Statement: The complement of the union of two sets is equal to the intersection of their complements.
Mathematical Representation: (A ∪ B)' = A' ∩ B'
Example:
Let U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3}, and B = {3, 4, 5}.
A ∪ B = {1, 2, 3, 4, 5}
(A ∪ B)' = {6}
A' = {4, 5, 6} and B' = {1, 2, 6}
A' ∩ B' = {6}
Therefore, (A ∪ B)' = A' ∩ B' = {6}.
De Morgan's Second Law
Statement: The complement of the intersection of two sets is equal to the union of their complements.
Mathematical Representation: (A ∩ B)' = A' ∪ B'
Example:
Let U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3}, and B = {3, 4, 5}.
A ∩ B = {3}
(A ∩ B)' = {1, 2, 4, 5, 6}
A' = {4, 5, 6} and B' = {1, 2, 6}
A' ∪ B' = {1, 2, 4, 5, 6}
Therefore, (A ∩ B)' = A' ∪ B' = {1, 2, 4, 5, 6}.
2. Distributive Law
Distributive Laws govern how operations of union and intersection distribute over each other.
Distributive Law of Union over Intersection
Statement: The union of a set with the intersection of two other sets is equal to the intersection of the union of the first set with each of the two other sets.
Mathematical Representation: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Example:
Let A = {1, 2}, B = {2, 3}, and C = {2, 4}.
B ∩ C = {2}
A ∪ (B ∩ C) = {1, 2}
A ∪ B = {1, 2, 3} and A ∪ C = {1, 2, 4}
(A ∪ B) ∩ (A ∪ C) = {1, 2}
Therefore, A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) = {1, 2}.
Distributive Law of Intersection over Union
Statement: The intersection of a set with the union of two other sets is equal to the union of the intersection of the first set with each of the two other sets.
Mathematical Representation: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Example:
Let A = {1, 2}, B = {2, 3}, and C = {2, 4}.
B ∪ C = {2, 3, 4}
A ∩ (B ∪ C) = {2}
A ∩ B = {2} and A ∩ C = {2}
(A ∩ B) ∪ (A ∩ C) = {2}
Therefore, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) = {2}.