De Morgan’s Theorem

DE MORGAN’S THEOREM

De Morgan’s First Theorem

The first theorem states that the complement of the sum of two logical inputs is equal to the product of its complements.

Proof

The Boolean equation for NOR gate is 

The Boolean equation for a bubbled AND gate is 

Both cases generate same outputs for same inputs. It can be verified using the following truth table.

From the above truth table, we can conclude .

Thus De Morgan’s First Theorem is proved. It also says that a NOR gate is equal to a bubbled AND gate.

The corresponding logic circuit diagram is shown in Figure 9.47.

De Morgan’s Second Theorem

The second theorem states that the complement of the product of two inputs is equal to the sum of its complements.

Proof

The Boolean equation for NAND gate is 

The Boolean equation for bubbled OR gate is 

A and B are the inputs and Y is the output. The above two equations produces the same output for the same inputs. It can be verified by using the truth table

From the above truth table we can conclude 

Thus De Morgan’s First Theorem is proved. It also says, a NAND gate is equal to a bubbled OR gate.

The corresponding logic circuit diagram is shown in Figure 9.48

EXAMPLE: 9 . 12

Simplify the Boolean identity

AC + ABC = AC

Solution

Step 1: AC (1 + B) = AC.1 [OR law-2]

Step 2: AC . 1 = AC [AND law – 2]

Therefore, AC + ABC = AC

Circuit Description

Thus the given statement is proved.