BOOLEAN ALGEBRAIC THEOREM

De Morgan’s Theorem

• These are very useful in simplifying expressions in which a product or sum of variables is inverted.
• De Morgan’s theorem represents two of the most important rules of Boolean algebra.

Transposition Theorem

Consensus Theorem/Redundancy Theorem

• This theorem is used to eliminate redundant term.
• A variable is associated with some variable and its compliment is associated with some other variable and the next term is formed by the left over variables, then the term becomes redundant.
• It is applicable only if a Boolean function,
  (i) Contains 3-variables.
  (ii) Each variable used two times.
  (iii) Only one variable is in complemented or uncomplemented form.
  Then, the related terms to that complemented and uncomplemented variable is the answer.
• Consensus theorem can be extended to any number of variables.

Duality Theorem

• It is one of the elegant theorems proved in advance mathematics.
• “Dual expression” is equivalent to write a negative logic of the given Boolean relation. For this we,
  (i) Change each OR sign by an AND sign and vice-versa.
  (ii) Compliment any ‘0’ or ‘1’ appearing in expression.
  (iii) Keep literals as it is.

Complementary Theorem

• For obtaining complement expression we
  (i) Change each OR sign by AND sign and vice-versa.
  (ii) Complement any ‘0’ or ‘1’ appearing in expression.
  (iii) Complement the individual literals.