The Laws of Logic: Boolean Algebra A State High Math Club Presentation START==TRUE

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The Laws of LogicBoolean AlgebraA State High
Math Club PresentationSTARTTRUE
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What is Boolean Algebra?

• Two values
• True and False - Logic Set Theory
• 1 and 0 - Computers Probability
• High and Low Digital Electronics

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AND - Basic

• Notation - AND
• (A AND B) returns true iff A and B are both true
• Numerically A B
• Identity - 1
• 0 AND 1 0
• 1 AND 1 1
• Annihilator - 0
• 0 AND 0 0
• 1 AND 0 0

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AND - Representations
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OR (Inclusive or) - Basic

• Notation - OR v
• (A OR B) returns true if either (or both) of A
  and B are true
• Numerically A B - A B
• Probability P(A or B) P(A)P(B)-P(A AND B)
• And/Or Construction
• Identity - 0
• Annihilator - 1

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OR - Representations
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NOT - Basic

• Notation NOT !
• Unary - only takes one argument
• Others are called binary
• Returns the opposite of its argument
• Analogous to negative sign

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NOT - Representations
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XOR (Exclusive or) - Derived

• Notation XOR
• (A XOR B) returns true iff exactly one of A and B
  is true
• Can also be considered not equals
• Either/or construction
• A XOR B (A B) v (A B)
• Numerically AB (mod 2)
• Also AB 2(AB) Same formula in probability
• A XOR 1 NOT(A)
• A XOR 0 A

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XOR - Representations
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Equivalence - Derived

• Notation
• Returns true iff AB

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Material Implication - Derived

• Notation xgty
• Linguistically If X, then Y
• X is called the antecedent y is called the
  consequent
• False ONLY when x is true and y is false
• NOT(X) OR Y
• More intuitively NOT(X AND NOT(Y))
• Why is x-gty true when x is false???
• This means that If two is odd, then two is even
  is a true statement!

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TautologiesThe Theorems of Boolean Algebra

• Called laws
• Proven with either truth tables or derivations
• Truth table true iff last column is all 1s
  (i.e. true for all input values)
• Example Proof of X X 0

X X XX XX 0
1 0 0 1
0 1 0 1
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De Morgans LawsDistributing a NOT

• (xVy)(x)(y)
• (xy)(x)V(y)
• Proofs?

X Y (x)(y) (xvy)
1 1 0 0
1 0 0 0
0 1 0 0
0 0 1 1
X Y (x)v(y) (xy)
1 1 0 0
1 0 1 1
0 1 1 1
0 0 1 1
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Important Laws

• AND and OR are
• Distributive over each other
• Associative
• Commutative
• X X
• XX0
• XvX1
• De Morgans Laws

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A Derivation

• (w V x) V (y V z) ((w V x) V y) V z (w V (x
  V y)) V z (w V (y V x)) V z ((w V y) V x) V
  z (w V y) V (x V z)
• Why is this valid?

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ApplicationsLogic Deduction

• Propositional Calculus gives valid forms of
  arguments
• Arguments are valid iff they are laws of boolean
  algebra
• Tends to use -gt a lot
• Examples
• (P (P-gtQ)) -gt Q Modus Ponens
• (Q (P-gtQ)) -gt P Modus Tollens
• ((P-gtQ) (Q-gtR)) -gt (P-gtR) Hypothetical
  Syllogism

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Modus Ponens Proof
P Q P-gtQ (P (P-gtQ)) (P (P-gtQ)) -gt Q
1 1 1 1 1
1 0 0 0 1
0 1 1 0 1
0 0 1 0 1
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Modus Tollens Proof
P Q P-gtQ (Q (P-gtQ)) (Q (P-gtQ)) -gt P
1 1 1 0 1
1 0 0 0 1
0 1 1 0 1
0 0 1 1 1
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Hypothetical Syllogism Proof
P Q R P-gtQ Q-gtR P-gtQ Q-gtR P-gtR
1 1 1 1 1 1 1
1 1 0 1 0 0 0
1 0 1 0 1 0 1
1 0 0 0 1 0 0
0 1 1 1 1 1 1
0 1 0 1 0 0 1
0 0 1 1 1 1 1
0 0 0 1 1 1 1
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ApplicationsComputer Curcuits

• Everything in a computer is either a 1 or 0
  (called a bit, or binary unit)
• 1 is high voltage 0 is low voltage
• Why?
• Calculations are done with AND, OR, and NOT logic
  gates
• Sound familiar?

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How Do Computers Add?

• We want to add two numbers, which are sequences
  of bits
• We add one place value (two bits) at a time
• Input The two bits, A and B
• Output sum bit (the actual place value) and
  carry bit (if we need to carry a 2)
• How can we make this circuit?

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Adding

• Make a truth table and translate it into a
  circuit diagram

A B Sum Carry
1 1 0 1
1 0 1 0
0 1 1 0
0 0 0 0
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Adding

• Sum A XOR B
• Carry A AND B

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What about the carry bit?

• We forgot we might need to add a carry bit as
  well
• So we really have three inputs A,B, and Cin
  (carry in)
• Still two outputs
• Cout (carry out)
• Sum

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Take Two
A B Cin S Cout
1 1 1 1 1
1 1 0 0 1
1 0 1 0 1
1 0 0 1 0
0 1 1 0 1
0 1 0 1 0
0 0 1 1 0
0 0 0 0 0

• How do we make this circuit diagram?

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Full 1-bit Adder
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How about more bits?
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How many basic gates?

• We use AND, OR, and NOT as basic gates
• Wouldnt it be nice to have ONE basic gate?
• Can mass-produce a single circuit dont have to
  worry about which (tiny and impossible to see)
  circuit is which

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NAND Sheffer Stroke

• Notation NAND ?
• A NAND B NOT (A AND B)
• Can be used to make AND, OR, and NOT gates
• How?

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NAND as a Universal Gate

• NOT(A) A NAND A
• A AND B NOT(A NAND B) (A NAND B) NAND (A NAND
  B)
• A OR B NOT(A) NAND NOT(B) by Demorgans Law
  (A NAND A) NAND (B NAND B)
• A -gt B NOT(A) OR B A NAND NOT(B) A NAND (B
  NAND B)

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NOR Pierce Arrow

• Notation NOR ?
• A NOR B NOT(A OR B)
• Can be used to make AND,OR, and NOT gates
• How?

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NOR as a Universal Gate

• NOT(A) A NOR A
• A OR B NOT(A OR B) (A NOR B) NOR (A NOR B)
• A AND B NOT(A) NOR NOT(B) by Demorgans Law
  (A NOR A) NOR (B NOR B)
• A-gtB NOT(A) OR B (NOT(A) NOR B) NOR (NOT(A)
  NOR B)
• ((A NOR A) NOR B) NOR ((A NOR A) NOR B)

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An Everyday ExampleSearch Engines

• Google uses boolean algebra on your search terms
• Automatically uses AND (i.e. whitespace AND)
• boolean algebra -gt Pages with BOTH boolean and
  algebra in them
• Use OR for logical OR
• Boolean OR algebra -gt Pages with EITHER boolean
  or algebra in them
• Use - for NOT
• Boolean Algebra -gtPages WITH boolean but
  WITHOUT algebra

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