Transcript Chapter 3 Logic Gates and Boolean Algebra

Chapter 3 Notes – Part II
Review Questions
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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.
NOT operation
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Truth table, Symbol, Sample waveform
Summary of Boolean Operations
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OR
0+0=0
AND
0•0=0
NOT
1’=0
0+1=1
1+0=1
1+1=1
0•1=0
1•0=0
1•1=1
0’=1
(NOTE THE SYMBOL USED FOR NOT!)
3-6 Describing logic circuits
algebraically
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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
Parentheses
(Often needed to establish precedence;
sometimes used optionally for clarity)
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How to interpret AB+C?
– Is it AB ORed with C ?
– Is it A ANDed with B+C ?
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 :
Circuits Contains INVERTERs
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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.
More Examples
Precedence
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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
Determining output level from a
diagram
Determine the output for the
condition where all inputs are LOW.
3-8 Implementing Circuits From
Boolean Expressions
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When the operation of a circuit is defined by a Boolean
expression, we can draw a logic-circuit diagram directly
from that expression.
Example
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Draw the circuit diagram to implement the expression
x  ( A  B )( B  C )
Review Question
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Draw the circuit diagram that implements the expression
x  A BC ( A  D ) Using gates having no more than three inputs.
3-9 NOR GATES AND NAND GATES
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NOR Symbol, Equivalent Circuit, Truth Table
Example
Example
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Determine the Boolean expression for a three-input NOR gate
followed by an INVERTER
NAND Gate
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Symbol, Equivalent circuit, truth table
Example
Example
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Implement the logic circuit that has the expression
using only NOR and NAND gates
x  AB  C  D 
Example
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Determine the output level in last example for A=B=C=1 and
D=0
Review Questions
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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 A=B=1, C=D=0
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?
3-10 Boolean Theorems
(single-variable)
Multivariable Theorems
x+y = y+x
xy = yx
commutativity
(x+y) + z = x + (y + z)
(xy)z = x(yz)
associativity
x(y+z) = xy + xz
x + yz = (x+y) (x+z) distributivity
x + xy = x
pf:
x+xy = x1 + xy = x(1+y) = x1 = x
Examples
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Simplify the expression
Simplify z   A  B  A  B 
Simplify x  ACD  A BCD
y  A BD  A B D
Review Questions
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Simplify
Simplify
Simplify
y  A C  AB C
y  A BC D  A B C D
y  A D  ABD
3-11 Demorgan’s Theorems
x  y  
xy
x  y  
x y
Example
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Simplify the expression z   A  C  B  D 
single variables inverted.
to one having only
Implications of DeMorgan’s
Theorems(I)
Implications of DeMorgan’s
Theorems(II)
Example
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Determine the output expression for the below circuit and
simplify it using DeMorgan’s Theorem
Review Questions
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Using DeMorgan’s Theorems to convert the expressions to one
that has only single-variable inversions.
z  A  B  C
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y  R ST  Q
Use only a NOR gate and an INVERTER to implement a circuit
having output expression:
z  A BC
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Use DeMorgan’s theorems to convert below expression to an
expression containg only single-variable inversions.
y  A  B  CD
3-12 Universality of NAND and
NOR gates
Universality of NOR gate
Example
Example
3-13 Alternate Logic-Gate
Representations
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Standard and alternate symbols for various logic gates and
inverter.
How to obtain the alternative
symbol from standard ones
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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)
Several points
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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.
Logic-symbol interpretation
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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 activehigh, otherwise it is active-low.
Interpretation of the two NAND
gate symbols
Interpretation of the two OR
gate symbols
Review Questions
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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