DeMorgan's Law
Worksheet

Many Boolean algebra expressions do not lend themselves to simplification using the simple Boolean algebra identities, e.g.,

DeMorgan's Law can help with expressions of this sort. It comes in two flavors, both of which are useful.

Proving DeMorgan's Law

Before we apply DeMorgan's Law, let us consider their proofs.

The proof that the first version of DeMorgan's Law, , can be ascertained from the following truth table. The two rightmost columns (in red) provide the truth values for the left and right sides of the equation, respectively. They are identical, proving the law.

INPUTS		OUTPUTS
A	B
0 0 1 1	0 1 0 1	1 1 0 0	1 0 1 0	0 0 0 1	1 1 1 0	1 1 1 0

To complete the proof of the second version of DeMorgan's Law, , fill in the following table.

INPUTS		OUTPUTS
A	B			A + B
0 0 1 1	0 1 0 1				1 0 0 0	1 0 0 0

Using DeMorgan's Law

To make use of DeMorgan's Law when simplifying Boolean expressions, follow the following steps.

1. Change all AND expressions to OR expressions, or vice versa.
2. Complement the individual terms that were ANDed or ORed.
3. Complement the whole expression.

Let us apply this procedure to . Note the use of the double-shafted arrow to indicate transformation, not equality.

1. Change OR to AND.
   =>
   
2. Complement each term.
   =>
   
3. Complement the entire expression.
   =>
   This final stage is indeed equal to amd can now be simplified using the Law of the Double Negative, yielding . We have thus converted the left side to the right side of DeMorgan's Law.

Here are three examples which show how DeMorgan's Law can be applied. The first two explicitly show the three steps described above, while the third presents two different simplifications of a single expression.

• => => => =
  
  
• => => => =
  
  
• 
  
  

  Solution I	Solution II
  DeMorgan's Law on whole expression = DeMorgan's Law on each expression = Associative Property = Commutative Property = Law of Tautology =	Commutative Property = Law of Tautology = DeMorgan's Law =

DeMorgan's Law Problems

DeMorgan's Law is obviously not useful in all situations. Typically, it is used only when a NOT bar covers all or part of an expression.

Here are a couple expressions to simplify. Then draw the logic circuit for the original Boolean expression and the resulting one. Finally, create the truth table for each to verify your work.