DeMorgan’s Theorems

DeMorgan, a mathematician who knew Boole, proposed two theorems that are an important part of Boolean algebra. In practical terms, DeMorgan’s theorems provide mathematical verification of the equivalency of the NAND and negative-OR gates and the equivalency of the NOR and negative-AND gates,

Apply DeMorgan’s theorems to the simplification of Boolean expressions

DeMorgan’s first theorem is stated as follows:

The complement of a product of variables is equal to the sum of the complements of the variables.

Stated another way:

The complement of two or more ANDed variables is equivalent to the OR of the complements of the individual variables.

The formula for expressing this theorem for two variables is:

DeMorgan’s second theorem is stated as follows:

The complement of a sum of variables is equal to the product of the complements of the variables.

Stated another way:

The complement of two or more ORed variables is equivalent to the AND of the complements of the individual variables.

The formula for expressing this theorem for two variables is:



Applying DeMorgan’s Theorems

 The following procedure illustrates the application of DeMorgan’s theorems and Boolean algebra to the specific expression: 


Step 1: Identify the terms to which you can apply DeMorgan’s theorems, and think of each term as a single variable.


Step 2: Since X + Y = XY,

Step 3: Use rule 9 (A = A) to cancel the double bars over the left term (this is not part of DeMorgan’s theorem).


Step 4: Apply DeMorgan’s theorem to the second term.


Step 5: Use rule 9 (A = A) to cancel the double bars over the E + F part of the term.

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