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 Theorem**De 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.