Section 10.3
Logic Gates
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Boolean algebra & circuits
• Boolean algebra is used to model electronic
circuitry
– Each input & output of electronic device can be
thought of as a member of the set {0,1}
– Electronic devices made up of circuits
– Each circuit can be designed using rules of
Boolean algebra
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Logic gates
• Basic elements of circuits are called gates
• Each type of gate implements a Boolean
operation (e.g. Boolean product, Boolean
sum, complement)
• Combinational circuits or gating networks:
– give output that depends only on input, not
current state of circuit
– have no memory capability
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Elements of combinational circuits
Inverter: accepts value of a Boolean variable as
input and produces the complement as output
OR gate: inputs are values of 2 or more
Boolean variables; output is Boolean sum of
their values
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Elements of combinational circuits
AND gate: inputs are values of 2 or more
Boolean variables; output is Boolean product of
their values
AND gates and OR gates may have multiple
inputs
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Combinations of gates
• Combinational circuits can be constructed
using combinations of inverters, OR gates,
and AND gates
• In such circuit combinations, some gates
may share inputs
• Output from a gate may be used as input by
another element
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Example 1
Construct a circuit to produce the output: ( x  y ) x
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Example 2
Construct a circuit to produce the output:
( x  y  z )( x y z )
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Example 3
• We can use circuits to model various types
of computational operations
• For example, suppose a committee of 3
people decides issues for an organization;
each one votes yes or no for each proposal
that arises, and a proposal passes if it
receives at least 2 yes votes
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Example 3
• Let x, y and z represent the voters’ decisions - a
value of 1 means a yes vote, and 0 means a no
vote
• We can design a circuit to output 1 when two
or more of x, y and z are 1 by representing the
Boolean function xy + xz + yz
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Example 3
The resulting circuit looks like this:
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Example 4
• Light fixtures may be controlled by more
than one switch
• Circuits for such fixtures must be designed
so that flipping any switch in the circuit will
turn the light on when it is off and off when
it is on
• We will look at such a circuit with 2
switches
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Example 4
• We represent the two switches as Boolean
variables x and y
• When a switch is open, its value is 0, and 1
when it is closed
• We need to define a function F(x,y)=1 when
the light is on, 0 when it is off
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Example 4
• Suppose the light is on if both switches are
closed; then F(1,1) = 1
• Opening either switch turns the light off, so
F(0,1) = F(1,0) = 0
• If one switch is open (and the light is off),
opening the other switch will turn the light
on, so F(0,0) = 1
• We can get these results if F(x,y) = xy + x y
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Example 4
A three-way circuit can be designed in a similar
fashion; for the three switches x,y,z the function
F(x,y,z) = xyz  x y z  x yz  x y z
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Adders
• Logic circuits can be used to carry out addition of
two positive integers from their binary expansions
• Such circuits form the basis of computer
arithmetic
• Half adder: a circuit that takes 2 bits as input,
outputting a sum bit and a carry bit; a half adder is
a multiple output circuit
• A full adder computes the sum and carry bits when
2 bits and a carry are added
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Half adder
Table below shows the possible inputs and corresponding
outputs of a half adder: x and y are the input bits, s is their
sum, and c is the carry:
x
1
1
0
0
y
1
0
1
0
s
0
1
1
0
c
1
0
0
0
From the table we can see that c = xy
and s = x y  x y
or ( x  y )( xy)
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Half adder
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Full adder
Inputs to full adder are the bits x and y and the carry ci; outputs
are the sum bit s and the new carry ci+1
x
1
1
1
1
0
0
0
0
y
1
1
0
0
1
1
0
0
ci
1
0
1
0
1
0
1
0
s
1
0
0
1
0
1
1
0
ci+1
1
1
1
0
1
0
0
0
The outputs of the full adder, which are
the sum bit and the new carry, are given
by the sum-of-products expansions:
Sum:
xyci  x y ci  x y ci  x yci
Carry: xyci  xyci  x yci  x yci
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Full adder
A full adder can be built using half adders to produce the
desired output:
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Section 10.3
Logic Gates
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