Chapter 4 Binary, Boolean Logic, and Gates

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Chapter 4 Binary, Boolean Logic, and Gates
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A Classic Joke....

• There are 10 types of people in the world, those
  who can read binary and those who cannot

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Two's Compliment Notation

• A way of representing signed numbers in a
  computer
• Algorithm
• Convert the number to binary
• Invert (flip) all bits
• Add 1

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Two's Compliment Example

• Convert the number -20 to its 8-bit binary two's
  complement form
• 2010 000101002
• 00010100 inverted is 11101011
• Adding 1 we get 11101100

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Two's Compliment

• One representation for 0
• 000000002 is 010
• 100000002 is -12810
• 1 bit for sign, 7 bits to represent data
• Thus what's the range of values?
• Allows subtraction by addition

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Boolean Logic

• Logical expressions that always evaluate to two
  values true and false

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Example

• Convert the phrase Adam is not a good instructor
  and Math is fun to a boolean expression

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Example (cont)

• Let a represent the phrase Adam is a good
  instructor
• Let b represent the phrase Math is fun
• Then the boolean expression is
• (NOT a) AND b
• Sometimes written a b

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Boolean Logic Operators

• Sometimes called Truth Functional Connectives
• a AND b
• True only when both a and b are true
• a OR b
• False only when both a and b are false
• NOT a
• Reverses the truth-value of a

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Precedence

• Some expressions are ambiguous
• NOT a AND b OR c
• Should this be interpreted as
• ((NOT a) AND b) OR c
• (NOT (a AND b)) OR c
• NOT ((a AND b) OR c)
• NOT (a AND (b OR c))

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Precedence (cont)

• Typically (in the absence of parenthesis)
• NOT is evaluated first
• AND is evaluated next
• OR is evaluated last
• So example would be
• ((NOT a) AND b) OR c

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Truth Tables

• Used to illustrate all the possible combinations
  of true/false values for a particular boolean
  expression
• Ex truth table for a AND b

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Gates

• Text defn An electronic device that operates on
  a collection of binary inputs to produces a
  binary output
• More concisely
• The electronic embodiment of boolean logic

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Gates
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Circuits

• Gates are the fundamental building blocks of
  circuits

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Circuit Example

• What is the equivalent boolean expression?