There are three basic operations in set theory - union, intersection and complement. De Morgan's Laws are all about relations among these set operations.
Let us recall these set operations first.1) Union: Union of two finite sets is defined as the set which has all the elements of both sets, such as A $\cup$ B = {x : x $\in$ A or x $\in$ B}
2) Intersection: Intersection of two finite sets denotes the set that contains the elements that are common to both the sets, such as A $\cap$ B = {x : x $\in$ A and x $\in$ B}
3) Complement: Complement of a finite set is the set of all the element except for the elements of that particular set. i.e., $\bar{A}$ or A$^{c}$ = {x : x $\notin$ A} = U - A, where A is the universal set.
De Morgan's Theorem defines two laws or rules which are as follows:First Law
It states that the complement of union of two finite sets is equal to the intersection of complements of the sets separately.Let us consider two sets P and Q, then according to De Morgan's law:$(A \cup B)^{c}=A^{c} \cap B^{c}$
Second Law
It states that the complement of intersection of two finite sets equals to the union of complements of the sets separately.
Let us consider two sets P and Q, then according to De Morgan's law:$(A \cap B)^{c}=A^{c} \cup B^{c}$
Generalization of De Morgan's TheoremDe Morgan's theorem is also generalized to be defined on n number of finite sets. Let us consider n finite sets A$_{1}$, A$_{2}$, A$_{3}$, ..., A$_{n}$.
Then generalization of De Morgan's theorem is given by the statement:
$(\bigcup_{i=1}^{n}A_{i})^{c}=\bigcap_{i=1}^{n}A_{i}^{c}$
And
$(\bigcap_{i=1}^{n}A_{i})^{c}=\bigcup_{i=1}^{n}A_{i}^{c}$
Undoubtedly, De Morgan's laws are the most important tools used while simplifying set operations and also while dealing with logic gates in computers.
