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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: xyx )(

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Example 2

Construct a circuit to produce the output:

))(( zyxzyx

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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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• 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 +

Example 4

yx

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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) = zyxzyxzyxxyz

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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 correspondingoutputs of a half adder: x and y are the input bits, s is theirsum, and c is the carry:

x y s c1 1 0 11 0 1 00 1 1 00 0 0 0

From the table we can see that c = xyand s = yxyx or ))(( xyyx

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Half adder

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Full adderInputs to full adder are the bits x and y and the carry ci; outputsare the sum bit s and the new carry ci+1

x y ci s ci+1

1 1 1 1 11 1 0 0 11 0 1 0 11 0 0 1 00 1 1 0 10 1 0 1 00 0 1 1 00 0 0 0 0

The outputs of the full adder, which arethe sum bit and the new carry, are givenby the sum-of-products expansions:

iiii cyxcyxcyxxyc Sum:

Carry: iiii ycxcyxcxyxyc

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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