Boolean De Morgan

De Morgan’s law

De Morgan’s law can help us simplify expressions which have lots of NOTs in them. It says that if we have a NOT of an AND, we can rewrite it as an OR of NOTs, and if we have a NOT of an OR, we can rewrite it as an AND of NOTs.

If you’d like a less wordy explanation of it, we ‘break the line and change the sign’, so, for example, \overline{A \cdot B} becomes \overline{A} + \overline{B}.

\overline{A \cdot B} = \overline{A} + \overline{B}
\overline{A + B} = \overline{A} \cdot \overline{B}

Like all booleans identities, we can use this in both directions, so we can also say that \overline{A} + \overline{B} = \overline{A \cdot B} and \overline{A} \cdot \overline{B} = \overline{A + B}.

Using it to simplify expressions

It’s most useful when we have an expression with two NOTs ‘on top’ of each other, for example, in this example:

Simplify \overline{\overline{A} \cdot \overline{B}}

• Use De Morgan’s law to break the line and change the sign:
  • \overline{\overline{A}} + \overline{\overline{B}}
• Now we have two NOTs on top of each other, so we can simplify them, using the fact that \overline{\overline{A}} = A (see [[ /boolean double negation|double negation]]):
  • A + B
• So the final, simplified expression is A + B.

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