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Section 10.3 Logic Gates

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**Boolean algebra & circuits**

4/17/2017 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 ch10.3

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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:

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Example 2 Construct a circuit to produce the output:

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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 +

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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) =

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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 y s c From the table we can see that c = xy and s = or

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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 y ci s ci+1 The outputs of the full adder, which are the sum bit and the new carry, are given by the sum-of-products expansions: Sum: Carry:

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Full adder A full adder can be built using half adders to produce the desired output:

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4/17/2017 Section 10.3 Logic Gates ch10.3

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