Combinational Logic and Boolean Algebra: Concepts, Theorems, and Applications, Slides of Digital Systems Design

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Combinational Logic

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Outlines

• Review of Boolean Algebra
• Specification
• Synthesis

Some Definitions

• Complement: variable with a bar over it

A , B , C

• Literal: variable or its complement

A , A , B , B , C , C

• Implicant: product of literals

ABC , AC , BC

• Minterm: product that includes all input variables

ABC , ABC , ABC

• Maxterm: sum that includes all input variables

(A+B+C) , (A+B+C) , (A+B+C)

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Digital Discipline: Binary Values

• Typically consider only two discrete values:
  • 1’s and 0’s
  • 1, TRUE, HIGH
  • 0, FALSE, LOW
• 1 and 0 can be represented by specific

voltage levels, rotating gears, fluid levels,

etc.

• Digital circuits usually depend on specific

voltage levels to represent 1 and 0

• Bit : B inary dig it

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Boolean Algebra

• Set of axioms and theorems to simplify Boolean equations
• Like regular algebra, but in some cases simpler because variables can have only two values (1 or 0)
• Axioms and theorems obey the principles of duality: - ANDs and ORs interchanged, 0’s and 1’s interchanged

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Boolean Axioms

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T1: Identity Theorem

1 =

=

B

0

B (^) B

B

• B 1 = B

• B + 0 = B

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T2: Null Element Theorem

• B 0 =

• B + 1 =

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T3: Idempotency Theorem

• B B =

• B + B =

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T3: Idempotency Theorem

• B B = B

• B + B = B

B =

B

B

B (^) B

B

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T4: Identity Theorem

• B = B

B = B

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T5: Complement Theorem

• B B =

• B + B =

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Boolean Theorems: Summary

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AND, OR, NOT A B C 0 0 0 0 1 0 1 0 0 1 1 1

AND

A 1 A

A B C 0 0 0 0 1 1 1 0 1 1 1 1

OR

A 1 1 A 0 0

A 0 A

A C 0 1 1 0

NOT

Review of Boolean algebra and switching functions

0 dominates in AND 1 dominates in OR