The two De Morgan theorem can be stated as below

  1. The complement of sum of the variables is equal to the product of complement of variables. \overline{A+B} = \overline{A}.\overline{B}
  2. The complement of product of the variables is equal to the sum of the complement of variables. \overline{A.B} = \overline{A} + \overline{B}

To verify this theorem let’s construct a truth table

12345678910
\overline{A} \overline{B} \overline{A+B} \overline{A+B} \overline{A} \overline{B} \overline{A}.\overline{B} \overline{A.B} \overline{A.B} \overline{A} + \overline{B}
0001111011
0110100011
1010010011
1110000100

If we compare column 4 and 7, they are identical proving first law stating \overline{A+B} = \overline{A}.\overline{B}

If we compare column 9 and 10, they are identical proving second law stating \overline{A.B} = \overline{A} + \overline{B}