Lecture 3. Boolean Algebra, Logic Gates

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Lecture 3. Boolean Algebra, Logic Gates
CS147
2x

• Prof. Sin-Min Lee
• Department of Computer Science

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Chapter Goals

• Boolean Algebra
• Identify the basic gates and describe the
  behavior of each
• Combine basic gates into circuits
• Describe the behavior of a gate or circuit using
  Boolean expressions, truth tables, and logic
  diagrams

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What is a gate?

• Combination of transistors that perform
• binary logic
• So called because one logic state enables
• or gates another logic state
• For each gate, the symbol, the truth table,
• and the formula are shown

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ltB , ,, 0,1gt Algebraic System Binary
operations , Unary operation
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Properties of Boolean Algebra
jasonm Redo table (p101)
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Computers

• There are three different, but equally powerful,
  notational methods for describing the behavior
  of gates and circuits
• Boolean expressions
• logic diagrams
• truth tables

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Boolean algebra

• Boolean algebra expressions in this algebraic
  notation are an elegant and powerful way to
  demonstrate the activity of electrical circuits

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Truth Table

• Logic diagram a graphical representation of a
  circuit
• Each type of gate is represented by a specific
  graphical symbol
• Truth table defines the function of a gate by
  listing all possible input combinations that the
  gate could encounter, and the corresponding output

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Gates

• Lets examine the processing of the following
  six types of gates
• NOT
• AND
• OR
• XOR
• NAND
• NOR

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NOT Gate

• A NOT gate accepts one input value and produces
  one output value

Figure 4.1 Various representations of a NOT gate
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NOT Gate

• By definition, if the input value for a NOT gate
  is 0, the output value is 1, and if the input
  value is 1, the output is 0
• A NOT gate is sometimes referred to as an
  inverter because it inverts the input value

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AND Gate

• An AND gate accepts two input signals
• If the two input values for an AND gate are both
  1, the output is 1 otherwise, the output is 0

Figure 4.2 Various representations of an AND gate
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OR Gate

• If the two input values are both 0, the output
  value is 0 otherwise, the output is 1

Figure 4.3 Various representations of a OR gate
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XOR Gate

• XOR, or exclusive OR, gate
• An XOR gate produces 0 if its two inputs are the
  same, and a 1 otherwise
• Note the difference between the XOR gate and the
  OR gate they differ only in one input situation
• When both input signals are 1, the OR gate
  produces a 1 and the XOR produces a 0

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XOR Gate
Figure 4.4 Various representations of an XOR gate
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NAND and NOR Gates

• The NAND and NOR gates are essentially the
  opposite of the AND and OR gates, respectively

Figure 4.5 Various representations of a NAND gate
Figure 4.6 Various representations of a NOR gate
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Gates with More Inputs

• Gates can be designed to accept three or more
  input values
• A three-input AND gate, for example, produces an
  output of 1 only if all input values are 1

Figure 4.7 Various representations of a
three-input AND gate
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3-Input And gate
A B C Y 0 0 0 0 0 0 1
0 0 1 0 0 0 1 1 0 1
0 0 0 1 0 1 0 1 1 0
0 1 1 1 1
Y A . B . C
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Constructing Gates

• A transistor is a device that acts, depending on
  the voltage level of an input signal, either as a
  wire that conducts electricity or as a resistor
  that blocks the flow of electricity
• A transistor has no moving parts, yet acts like
  a switch
• It is made of a semiconductor material, which is
  neither a particularly good conductor of
  electricity, such as copper, nor a particularly
  good insulator, such as rubber

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Circuits

• Two general categories
• In a combinational circuit, the input values
  explicitly determine the output
• In a sequential circuit, the output is a function
  of the input values as well as the existing state
  of the circuit
• As with gates, we can describe the operations of
  entire circuits using three notations
• Boolean expressions
• logic diagrams
• truth tables

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Combinational Circuits

• Gates are combined into circuits by using the
  output of one gate as the input for another

AND
OR
AND
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Combinational Circuits
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(p100)
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• Because there are three inputs to this circuit,
  eight rows are required to describe all possible
  input combinations
• This same circuit using Boolean algebra
• (AB AC)

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Now lets go the other way lets take a Boolean
expression and draw
jasonm Redo table to get white space (p101)

• Consider the following Boolean expression A(B
  C)

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• Now compare the final result column in this truth
  table to the truth table for the previous example
• They are identical

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Simple design problem

• A calculation has been done and its results
• are stored in a 3-bit number
• Check that the result is negative by anding
• the result with the binary mask 100
• Hint a mask is a value that is anded with
• a value and leaves only the important bit

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Now lets go the other way lets take a Boolean
expression and draw

• We have therefore just demonstrated circuit
  equivalence
• That is, both circuits produce the exact same
  output for each input value combination
• Boolean algebra allows us to apply provable
  mathematical principles to help us design
  logical circuits

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Adders

• At the digital logic level, addition is performed
  in binary
• Addition operations are carried out by special
  circuits called, appropriately, adders

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Adders

• The result of adding two binary digits could
  produce a carry value
• Recall that 1 1 10 in base two
• A circuit that computes the sum of two bits and
  produces the correct carry bit is called a half
  adder
• Notice the Sum Carry are NEVER both 1.

(XOR)
(AND)
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Adders

• Circuit diagram representing a half adder
• Two Boolean expressions
• sum A ? B
• carry AB

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Adders

• A circuit called a full adder takes the carry-in
  value into account

Figure 4.10 A full adder
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Adding Many Bits

• To add 2 8-bit values, we can duplicate a
  full-adder circuit 8 times. The carry-out from
  one place value is used as the carry in for the
  next place value. The value of the carry-in for
  the rightmost position is assumed to be zero, and
  the carry-out of the leftmost bit position is
  discarded (potentially creating an overflow
  error).

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Universal Gates
How to use NOR gate to build a NOT gate?
Truth Table
A B C Q
0 0 0 1
1 1 1 0
Logic Gates
Hint! Link inputs B C together (to a same
source).
When A 0, B C A 0 When A 1, B C A
1
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Universal Gates
How to use NOR gates to build an OR gate?
Truth Table
A B C D E Q
0 0 1 1 1 0
0 1 0 0 0 1
1 0 0 0 0 1
1 1 0 0 0 1
NOT
NOR
D
A
C
Q
B
E
Hint 1 Use 2 NOR gates
Hint 2 From a NOR gate, build a NOT gate
Hint 3 Put this NOT gate after a NOR gate
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Universal Gates
How to use NOR gates to build an AND gate?
Truth Table
A B C D Q
0 0 1 1 0
0 1 1 0 0
1 0 0 1 0
1 1 0 0 1
Hint 1 Use 3 NOR gates
Hint 2 From 2 NOR gates, build 2 NOT gates
Hint 3 Each NOT gate is an
input to the 3rd NOR gate
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Universal Gates
How to use NOR gates to build a NAND gate?
Truth Table
A B C D E Q
0 0 1 1 0 1
0 1 1 0 0 1
1 0 0 1 0 1
1 1 0 0 1 0
Hint 1 Use 4 NOR gates
Hint 2 Use 3 NOR gates to build
a NAND gate (previous lesson)
Hint 3 Use the 4th NOR gate to build a NOT gate
Hint 4 Insert NOT gate after NAND gate
Hint 5 NOT-NAND AND
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Universal Gates
How to use NAND gates to build a NOT gate?
Truth Table
A B C Q
0 0 0 1
1 1 1 0
Logic Gates
Hint! Link inputs B C together (to a same
source).
When A 0, B C A 0 When A 1, B C A
1
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Universal Gates
How to use NAND gates to build an AND gate?
Truth Table
A B C Q
0 0 1 0
0 1 1 0
1 0 1 0
1 1 0 1
NOT
NAND
C
A
Q
B
Hint 1 Use 2 NAND gates
Logic Gates
Hint 2 From a NAND gate, build a NOT gate
Hint 3 Put this NOT gate after a NAND gate
Hint 4 NOT-NAND AND
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Universal Gates
How to use NAND gates to build an OR gate?
Truth Table
A B C D Q
0 0 1 1 0
0 1 1 0 1
1 0 0 1 1
1 1 0 0 1
Hint 1 Use 3 NAND gates
Logic Gates
Hint 2 Use 2 NAND gates to build 2 NOT gates
Hint 3 Put the 3rd NAND gate
after the 2 NOT gates
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Universal Gates
How to use NAND gates to build a NOR gate?
Truth Table
A B C D E Q
0 0 1 1 0 1
0 1 1 0 1 0
1 0 0 1 1 0
1 1 0 0 1 0
Hint 1 Use 4 NAND gates
Logic Gates
Hint 2 Use 3 NAND gates to build an OR gate
Hint 3 Use a NOR gate to build a NOT gate
Hint 4 Put the NOT gate after OR gate
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as Universal Logic Gates
NAND and NOR

• Any logic circuit can be built using only NAND
  gates, or only NOR gates. They are the only
  logic gate needed.
• Here are the NAND equivalents

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NAND and NOR as Universal Logic Gates (cont)

• Here are the NOR equivalents
• NAND and NOR can be used to reduce the number of
  required gates in a circuit.

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Practice Assignment (check the result)