Boolean Algebra

1
Boolean Algebra

• Digital Logic
• ELET 1133

2
Boolean Algebra

• after mathematician George Boole
• A means of reducing the number of logic gates
  needed to realize the desired output.

3
Logic Reduction Example

• Suppose you can survive one test in a day, but
  hit the PANIC BUTTON when there is a digital
  logic test and another test the same day.
• P Panic Button
• C Circuits Test.
• D Digital Logic Test.
• F Fabrication Test.

4
Boolean Equation

• P D and C or D and F
• Boolean Form P DC DF
• P D (CF)
• Draw logic gate diagram
• Three gates
• two AND gates
• followed by one OR gate

5
Boolean Reduction

• Look again
• P C and either H or R.
• Boolean P D(CF)
• Draw logic diagram
• Two gates
• one OR gate
• followed by one AND gate.

6
Drill Motor Boolean Equations

• CR1Stop (Start CR1)
• CR2CR1LS1LS2
• CR3LS2 (CR2CR3)
• Extend CR3
• Retract CR3
• Motor CR3

7
Boolean Algebra

• OR operation is Boolean ADDITION
• X A B
• AND operation is Boolean MULTIPLICATION
• X AB
• These operations follow some of the same laws as
  standard algebra.

8
Commutative Laws

• commute means exchange.
• mathematically exchange order gt same result.
• Commutative Law of Addition
• X A B B A
• Commutative Law of Multiplication
• X AB BA

9
Associative Laws

• Associate group together
• Addition
• X A (B C) (A B) C
• Multiplication
• X A(BC) (AB)C

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Distributive Laws

• Distribute divide into groups
• X A(B C) AB AC
• X (A B) (C D) AC AD BC BD
• changes logic from
• OR then AND to AND then OR
• Distributive law was used for reduction in panic
  example.

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Boolean Theorems

• Text calls Boolean Rules

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Operations with 1 and 0

• A0 0 (A and Zero is Zero)
• any 0 input gives 0 output with AND
• A1 A (A and One is A)
• A 0 A
• A 1 1

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Idempotent Laws

• Idem same
• AA A
• 0 and 0 0
• 1 and 1 1
• A A A
• 0 or 0 0
• 1 or 1 1

14
Define Compliment

• Compliment means inverse.
• Put A into a NOT gate and the result is
  A-compliment
• Sometimes written A
• rather than A with a bar over it
• apostrophe is easier to type.

15
Involution Law

• A double negative is a positive.
• Compliment the input twice and its back to the
  original.
• (A)A

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Laws of Complimentarity

• Combine input and inputs compliment.
• AA 0.
• Either the number or its compliment is always
  zero.
• A A 1.
• One term will always be zero.
• The other term will always be one.

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Simplification Theorems

• A AB A B
• A AB A B
• Cant derive from previous theorems, but can
  demonstrate the theorems validity with a truth
  table.
• Last two columns are the same. Thus the
  solution is the same for all possible A, B.

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More Simplification Theorems

• Can derive these from previous theorems.
• AB AB A
• (A B) (A B) A
• A AB A
• A(AB) A

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Proof of A AB A
AAB A(1B) distributive A(1B) A(1)
operations with 1, 0. A(1) A operations with
1, 0.