Lecture 5 EGRE 254

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Lecture 5EGRE 254

• 1/28/09

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Boolean algebra

• a.k.a. switching algebra
• deals with Boolean values -- 0, 1
• Positive-logic convention
• analog voltages LOW, HIGH --gt 0, 1
• Negative logic -- seldom used
• Signal values denoted by variables(X, Y, FRED,
  etc.)

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Boolean operators

• Complement X or (opposite of X)
• AND X Y
• OR X Y

binary operators, describedfunctionally by truth
table.
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More definitions

• Literal a variable or its complement
• X, X, FRED, CS_L
• Expression literals combined by AND, OR,
  parentheses, complementation
• XY
• P Q R
• A B C
• ((FRED Z) CS_L A B C Q5) RESET
• Equation Variable expression
• P ((FRED Z) CS_L A B C Q5)
  RESET

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Logic symbols
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Basic Axioms
A1 A1
A2 A2
A3 A3
A4 A4
A5 A5
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Proving theorems

• Using axioms or theorems already proven.
• Perfect induction Verify theorem for all
  possible values of the variables.
• 1 variable 2 21 possible values. 0, 1
• 2 variables 4 22 possible values. 00, 01, 10,
  11
• 3 variables 8 23 possible values. 000, 001, ,
  111
• n variables 2n possible values.
• For general case of n variable we use the
  mathematical technique of finite induction.

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Prove T1 and T1

• T1 X 0 X
• Proof 1a
• If X 0 then X 0 X by A4
• If X 1 then X 0 X by A5
• Proof 2a,b

• T1 X?1 X
• Proof 1b
• If X 1 then X?1 X by A4
• If X 0 then X?1 X by A5
• Proof 3b
• T1 follows from duality of T1.

X X0 X?1
0 0 0
1 1 1
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Basic Theorems
T1 T1
T2 T2
T3 Idempotent law T3
T4 T4 Same as T4
T5 T5
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Theorems
T6 Commutative law T6
T7 Associative law T7
T8 Distributive law T8
T9 Adsorption law T9
T10 T10
T11 T11
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T8

• Not what we would expect!
• Proof 1 using truth table (perfect induction)

X Y Z XY XZ YZ (XY)(XZ) X YZ
0 0 0 0 0 0 0 0
0 0 1 0 1 0 0 0
0 1 0 1 0 0 0 0
0 1 1 1 1 1 1 1
1 0 0 1 1 0 1 1
1 0 1 1 1 0 1 1
1 1 0 1 1 0 1 1
1 1 1 1 1 1 1 1
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T8

• Proof 2 Algebraically using proved theorems
• (X Y)(X Z) (XY)X (XY)Z Why?
• XXYXXZYZ
  T6, T8
• XXYXZYZ
  T3, T6
• X?1 X(YZ) YZ
  T1, T8
• X(1(YZ)) YZ
  T8
• X?1 YZ
  T6, T1
• X YZ
  T1
• Better
• (X Y)(X Z) X XZ XY YZ
• X(1ZY) YZ X YZ
• Proof 3 Follows from T8 and duality.

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Algebraic Proofs

• T10 XYXY X(YY) X?1 X
• T10 (XY)(XY) XXYXYYY
• X(1YY) 0 X(1) X
• T11 XYXZYZ XYXZ(XYZXYZ)
• XY(1Z) XZ(1Z) XY XZ
• T11 Do as an exercise.

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Example using T9

• (AB)C (AB)CD(EF) (AB)C
• Treat (AB)C as X, treat D(EF) as Y
• Or instead of using T9 recognize that
• (AB)C (AB)CD(EF) (AB)C(1D(EF))
  (AB)C
• It is not necessary to memorize all of these
  theorems.
• Know through T5 and couple that with your
  knowledge of ordinary algebra.

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XOR

• X ? Y XY XY
• X ? 0 X
• X ? 1 X
• X ? X 0
• X ? X 1
• X ? Y ? Z X ? (Y ? Z) Z ? X ?Y

X Y X?Y
0 0 0
0 1 1
1 0 1
1 1 0
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How are these XOR gates used?
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DeMorgans Theorem

• These are the equations you must memorize
• But notice that given one it is trivial to obtain
  the others.

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Prove
X Y X Y XY XY (XY)
0 0 1 1 1 0 1
0 1 1 0 0 1 0
1 0 0 1 0 1 0
1 1 0 0 0 1 0
Alternative proof. Let X 0 then 1Y (0
Y) Let X 1, then 0Y (1Y) 1 0
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DeMorgans Theorem in n variables
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Generalizations

• DeMorgans Theorem
• Duality. If
• then

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Shannons expansion theorem
Proof Consider f(xi) xif(0)xi xif(1)xi When
xi 0 then f(0)xi 1f(0)xi 0f(1)xi
f(0)xi When xi 1 then f(1)xi 0f(0)xi
1f(1)xi f(1)xi Thus, by perfect induction f(xi)
xif(0)xi xif(1)xi
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Implementation example

• Draw circuit directly from equations.
• Draw circuit using only NAND gates.

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Design example

• Design a 3-input majority circuit

X Y Z F
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
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Design example

• Design a 3-input majority circuit

X Y Z F
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
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Design example

• Design a 3-input majority circuit

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Design example

• Design a 3-input majority circuit

X Y Z F
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
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Design example

• Design a 3-input majority circuit

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Example

• Show how to build an 8 input and gate using
  several two input and gates.
• Which is better? Why?

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Schmitt-trigger gates

• contain input hysteresis. Useful for interfacing
  to slow or noisy signals.

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Tri-state buffers