De Morgans Theorems

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De Morgans Theorems

• Universal Gates

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De Morgans Theorem

• For NAND and NOR gates
• Split inverse terms
• Change OR to AND
• Change AND to OR
• Note (AB) does not equal AB

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DeMorgan's Theorem NAND

• A NAND gate is the same as inverters and an OR
  gate.

A
X
NAND
B
Simplified Symbol
OR
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DeMorgan's Theorem NOR

• A NOR is equivalent to an inverted input AND

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Bubble Pushing

• Method of changing logic gate using DeMorgans
  Theorem
• Change OR to AND
• Change AND to OR
• Put inversion bubbles anyplace there were none
• Remove inversion bubbles present.

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Example 5-17

• Reduce the equation

Associative DeMorgan's DeMorgan's Distributive Com
pli-mentarity
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Example 5-19a
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Gate Conversions

• Universality of NAND and NOR Gates

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Universality of NAND Gates

• From a NAND gate can make an inverter.
• tie both inputs to A

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AND Gate from NAND Gates

• NAND output
• then add a NAND inverter

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OR Gate from NAND Gates

• (AB) A B by DeMorgan's
• Let C A and D B
• Thus C A and D B
• Substituting
• (CD) C D
• By inverting the inputs to the NAND gate the
  result is an OR gate.
• Could also bubble push.

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OR Gate from NAND Gates
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Universality of NAND Gates

• Make a NOR gate
• Use OR gate and Inverter.

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Universality of NOR Gates

• Similarly can do the same thing with NOR gates.

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Example 5-19 c