De Morgan's Laws

The two theorems that simplify all of Boolean Algebra.

The Mnemonic

"Break the bar, change the sign."

This simple phrase summarizes both laws perfectly. If you have a large bar (NOT) over an expression, you can break it into smaller bars over individual variables and flip the sign (AND to OR, OR to AND).

Theorem 1: The NAND Equivalent

The complement of a product (NAND) is equal to the sum of the complements (Negative-OR).

A • B = A + B

In logic gate terms: A NAND gate is equivalent to an OR gate with inverted inputs.

Theorem 2: The NOR Equivalent

The complement of a sum (NOR) is equal to the product of the complements (Negative-AND).

A + B = A • B

In logic gate terms: A NOR gate is equivalent to an AND gate with inverted inputs.

Why is this useful?

  • Simplification: Reduce complex boolean expressions to fewer terms.
  • Universal Gates: It proves that you can build ANY digital circuit using only NAND gates or only NOR gates. This is cheaper for manufacturing.