Augustus De Morgan (1806-1871) was an Englishman, born in India, who became a mathematician and was co-founder of the London Mathematical Society. Among his several contributions to mathematics, he made rigorous the principle of induction, but is now best remembered for the two laws which bear his name. These laws are extremely useful when manipulating Boolean expressions, and are :-

The complement of a logical sum of terms is equivalent to the logical product of the complements of those terms, that is NOT ( A OR B OR ··· OR N ) = ( NOT A ) AND ( NOT B ) AND ··· AND ( NOT N ).

The complement of a logical product of terms is equivalent to the logical sum of the complements of those terms, that is NOT ( A AND B AND ··· AND N ) = ( NOT A ) OR ( NOT B ) OR ··· OR ( NOT N ).

Both are easily proven, from the basic definitions of a sum and a product.

A sum of terms, that is the logical OR of the those terms, is true if any term is true, and is false only when all terms are false. The complement of a sum is then true only when all terms are false, that is when the complement of every term is true, in other words when the product of complements is true. But that product is then false when any complement is false, that is when any term is true. This establishes the law that the complement of a sum of terms is equivalent to the product of the complemented terms.

A product of terms, that is the logical AND of those terms, is true only when every term is true, and is false if any term is false. The complement of a product is then true when any term is false, that is when the complement of any term is true, in other words when the sum of the complements is true. But that sum is then false only when all complements are false, that is when all terms are true. This establishes the law that the complement of a product of terms is equivalent to the sum of the complemented terms.

Both laws can be expressed differently, in a form which gives meaning to the use of a double negative. In these paraphrasings of de Morgan's laws, the word denial is used in place of the word complement, and untrue is used in place of false.

The logical OR of several possibilities is the denial of the logical AND of the denials of those possibilities. In other words, at least one possibility is asserted as true when it is denied that all are untrue, or in terms of Boolean functions A OR B OR ··· OR N = NOT ( ( NOT A ) AND ( NOT B ) AND ··· AND ( NOT N ) ).

The logical AND of several possibilities is the denial of the logical OR of the denials of those possibilities. In other words, all possibilities are asserted if it must be denied that any are untrue, or in terms of Boolean functions
A AND B AND ··· AND N = NOT ( ( NOT A ) OR ( NOT B ) OR ··· OR ( NOT N ) )