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Boolean Algebra Math & DeMorgan’s Law

03/13/2024

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Properties of Boolean Algebra

  • Boolean Algebra has commutative, associative and distributive properties

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  • DeMorgan’s Law is the most important property to know

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A+B = B+A

A * B = B * A

Commutative Property

(A+B)+C = A+(B+C)

(A*B)*C = A*(B*C)

Associative Property

A*(B+C) = A*B + A*C

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Distributive Property

NOT (A OR B) = (NOT A AND NOT B) = (~A * ~ B)

DeMorgan’s Law

NOT (A AND B) = (NOT A OR NOT B) = (~ A + ~ B)

DeMorgan’s Law

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DeMorgan’s Law Truth Table

***** Notice: Columns 4 & 7 are equal, and Columns 9 & 10 are equal *****

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More (Obvious) Properties

A OR 0 = A + 0 = A

A AND 0 = A * 0 = 0

A OR 1 = A +1 = 1

A AND 1 = A* 1 = A

A OR NOT A = A + ~ A = 1

A AND NOT A = A * ~ A = 0

A OR A = A + A= A

A AND A = A * A = A

NOT (NOT A) = ~ (~A) = A

~~A = A

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Equivalent Expressions

  • Two boolean expressions are equivalent when they produce the exact same truth table for all possible inputs
  • Two ways to find if two expressions are equivalent : truth tables & simplification

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  • Therefore, no matter the values of A & B, this expression is always true

called a tautology

    • A + 1 and A + ~A

NOT (A AND B) OR B

~ (A * B) + B

(NOT A OR NOT B) OR B

(~A + ~B) +B

NOT A OR (NOT B OR B)

~A + (~B +B)

NOT A OR 1

~ A+1

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Apply DeMorgan's law

Apply Associative property

Apply Obvious property

Apply OR property

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Reminders from Last Week

  • Boolean Algebra follows PEMDAS
    • Therefore [AND] is before [OR] - (* before +)
    • Boolean = True (1) (ON) OR False (0) (OFF)
    • Test for this topic will be made available on Wednesday 04/03/2024 due on 04/07
      • Test will be like last time on the computer
      • You will have 30 minutes to do 5 problems
      • That is 6 minutes per problem
  • Have fun – Practice make Perfect

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