Chapter 3 Notes Part II Review Questions

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Chapter 3 Notes Part IIReview Questions

• What is the only input combination that will
  produce a HIGH at the output of a five-input AND
  gate?
• What logic level should be applied to the second
  input of a two-input AND gate if the logic signal
  at the first input is to be inhibited(prevented)
  from reaching the output?
• True or false An AND gate output will always
  differ from an OR gate output for the same input
  conditions.

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NOT operation

• Truth table, Symbol, Sample waveform

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Summary of Boolean Operations

• OR
• 000 011 101 111
• AND
• 000 010 100 111
• NOT
• 10 01 (NOTE THE SYMBOL USED FOR NOT!)

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3-6 Describing logic circuits algebraically

• Any logic circuit, no matter how complex, can be
  completely described using the three basic
  Boolean operations OR, AND, NOT.
• Example logic circuit with its Boolean
  expression

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Parentheses (Often needed to establish
precedence sometimes used optionally for
clarity)

• How to interpret A?BC?
• Is it A?B ORed with C ?
• Is it A ANDed with BC ?
• Order of precedence for Boolean algebra AND
  before OR. Parentheses make the expression
  clearer, but they are not needed for the case on
  the preceding slide.
• Note that parentheses are needed here

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Circuits Contains INVERTERs

• Whenever an INVERTER is present in a
  logic-circuit diagram, its output expression is
  simply equal to the input expression with a bar
  over it.

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More Examples
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Precedence

• First, perform all inversions of single terms
• Perform all operations with paretheses
• Perform an AND operation before an OR operation
  unless parentheses indicate otherwise
• If an expression has a bar over it, perform the
  operations inside the expression first and then
  invert the result

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Determining output level from a diagram
Determine the output for the condition where all
inputs are LOW.
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3-8 Implementing Circuits From Boolean Expressions

• When the operation of a circuit is defined by a
  Boolean expression, we can draw a logic-circuit
  diagram directly from that expression.

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Example

• Draw the circuit diagram to implement the
  expression

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Review Question

• Draw the circuit diagram that implements the
  expression
• Using gates having no more than three inputs.

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3-9 NOR GATES AND NAND GATES

• NOR Symbol, Equivalent Circuit, Truth Table

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Example
15
Example

• Determine the Boolean expression for a
  three-input NOR gate followed by an INVERTER

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NAND Gate

• Symbol, Equivalent circuit, truth table

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Example
18
Example

• Implement the logic circuit that has the
  expression
• using only NOR and NAND gates

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Example

• Determine the output level in last example for
  ABC1 and D0

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Review Questions

• What is the only set of input conditions that
  will produce a HIGH output from a three-input NOR
  gate?
• Determine the output level in last example for
  AB1, CD0
• Change the NOR gate at last example to a NAND
  gate, and change the NAND to a NOR. What is the
  new expression for x?

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3-10 Boolean Theorems(single-variable)
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Multivariable Theorems
xy yx xy yx commutativity (xy) z
x (y z) (xy)z x(yz) associativity x(yz)
xy xz x yz (xy) (xz) distributivity x
xy x pf xxy x1 xy x(1y) x1
x
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Examples

• Simplify the expression
• Simplify
• Simplify

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Review Questions

• Simplify
• Simplify
• Simplify

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3-11 Demorgans Theorems
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Example

• Simplify the expression to one having
  only single variables inverted.

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Implications of DeMorgans Theorems(I)
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Implications of DeMorgans Theorems(II)
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Example

• Determine the output expression for the below
  circuit and simplify it using DeMorgans Theorem

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Review Questions

• Using DeMorgans Theorems to convert the
  expressions to one that has only single-variable
  inversions.
• Use only a NOR gate and an INVERTER to implement
  a circuit having output expression
• Use DeMorgans theorems to convert below
  expression to an expression containg only
  single-variable inversions.

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3-12 Universality of NAND and NOR gates
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Universality of NOR gate
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Example
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Example
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3-13 Alternate Logic-Gate Representations

• Standard and alternate symbols for various logic
  gates and inverter.

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How to obtain the alternative symbol from
standard ones

• Invert each input and output of the standard
  symbol, This is done by adding bubbles(small
  circles) on input and output lines that do not
  have bubbles and by removing bubbles that are
  already there.
• Change the operation symbol from AND to OR, or
  from OR to AND.(In the special case of the
  INVERTER, the operation symbol is not changed)

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Several points

• The equivalences can be extended to gates with
  any number of inputs.
• None of the standard symbols have bubbles on
  their inputs, and all the alternate symbols do.
• The standard and alternate symbols for each gate
  represent the same physical circuit there is no
  difference in the circuits represented by the two
  symbols.
• NAND and NOR gates are inverting gates, and so
  both the standard and the alternate symbols for
  each will have a bubble on either the input or
  the output, AND and OR gates are noninverting
  gates, and so the alternate symbols for each will
  have bubbles on both inputs and output.

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Logic-symbol interpretation

• Active high/low
• When an input or output line on a logic circuit
  symbol has no bubble on it, that line is said to
  be active-high, otherwise it is active-low.

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Interpretation of the two NAND gate symbols
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Interpretation of the two OR gate symbols
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Review Questions

• Write the interpretation of the operation
  performed by the below gate symbols
• Standard NOR gate symbol
• Alternate NOR gate symbol
• Alternate AND gate symbol
• Standard AND gate symbol