De Morgan’s Law
De Morgan's Law is a statement of transformation rules used in set theory. De Morgan's Law relates the intersection and union of sets through complements in two ways. These are primarily used to reduce expressions into a simpler form. According to the De Morgan’s Law,
(i) The negation of a disjunction is the conjunction of the negations.
¬(A V B)↔(¬A) Λ (¬B)
(ii) The negation of a conjunction is the disjunction of the negations.
¬(A Λ B)↔(¬A) V (¬B)
or
(i) The complement (c) of the union of two sets is the same as the intersection of their complements.
〖(A∪B)〗^c=A^c∩B^c
(ii) The complement of the intersection of two sets is the same as the union of their complements.
〖(A∩B)〗^c=A^c∪B^c
or
not (A or B) = (not A) and (not B)
(A∪B) ̅=A ̅∩B ̅
Not (A and B) = (not A) or (not B)
(A∩B) ̅=A ̅∪B ̅
Numerical Example:
A = {1, 2, 3}, B = {3, 2, 4} and U = {1, 2, 3, 4, 5, 6}
Here, U is the universal set of A and B
Now A∪B={1,2,3,4} and 〖 (A∪B)〗^c={5,6}
〖 A〗^c={4,5,6},〖 B〗^c={1,5,6} and〖 A〗^c∩B^c={5,6}
Therefore, 〖(A∪B)〗^c=A^c∩B^c={5,6}
On the other hand, A∩B={ 2,3,} and 〖 (A∩B)〗^c={1,4,5,6}
〖 A〗^c={4,5,6},〖 B〗^c={1,5,6} and〖 A〗^c∪B^c={1,4,5,6}
Therefore, 〖(A∩B)〗^c=A^c∪B^c={1,4,5,6}
Thus we have proved that
〖(A∪B)〗^c=A^c∩B^c and 〖(A∩B)〗^c=A^c∪B^c
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