# Points Addressed in this Lecture

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1 Points Addressed in this Lecture Lecture 3: Basic Logic Gates & Boolean Expressions Professor Peter Cheung Department of EEE, Imperial College London (Floyd , 4.1) (Tocci ) What are the basic logic gates? What is Boolean algebra? Boolean variables & expressions Boolean algebra as a way to write down logic Boolean Operators s Relationships between logic gates & Boolean expressions 3.1 Boolean Algebra Digital electronic systems manipulate binary information To design such systems we need a convenient mathematical framework useful systems are often too complicated to keep in our head Boolean algebra provides this framework Points in a circuit are represented by Boolean Variables Boolean algebra allows us to specify relationships between Boolean variables Hence Boolean algebra can be used as a design tool for digital electronic circuits Boolean Variables Boolean variables take the value either 0 or 1 only if a variable doesn't have the value 0, then it must have the value 1 if a variable doesn't have the value 1, then it must have the value 0 In digital electronics: Boolean 0 and 1 correspond to the binary 0 and 1 In logic: 1 and 0 are sometimes called true and false We use symbols to represent Boolean variables just like with ordinary algebra eg: A, B, C, X, Y, Z, etc typically a single character typically upper case Three Logic operations: AND, OR, NOT

2 Boolean Algebra to Describe Logic Example: House Heating System Principles: set the required temperature using a thermostat turn on heating if temperature lower than required turn off heating if temperature higher than required turn on heating if heating pipes are in danger of freezing Implementation: use a manual switch to turn on the house heating use a room thermostat to detect room temperature use a frost thermostat to detect outside temperature (danger of freezing) use a digital electronic circuit to turn the heating on and off 'intelligently' Boolean representation: 4 variables H, R, F and S H represents the On/Off switch of the entire heating system H = 1 when the heating system is switched on. R represents the room thermostat R = 1 when the room temperature is lower than required F represents the frost thermostat F = 1 when the external temperature is near freezing S represents the On/Off switch of the boiler S = 1 when heat should be generated by the boiler S should be 1 when (H=1 and R=1) or when (F=1 and R=1) In Boolean algebra we use for 'and' and + for 'or' S = H R + F R If we could build an electronic circuit which implemented this we could sell it as a simple heating system controller Boolean Operators Like ordinary algebra, Boolean algebra allows for operations on its variables NOT - Takes the complement (inverse) of a single variable Called 'NOT K' and written K eg: Let K represent a key on a computer keyboard and let K = 1 mean the key is pressed We now have a variable which shows the state of the key: K=1 shows key is pressed K=0 shows key is not pressed If we take the compliment of K we have a variable which also shows the state of the key but in the opposite sense K=1 shows key is not pressed K=0 shows is pressed

3 Basic Boolean Operators & Logic Gates Truth Tables Inverter How a logic circuit s output depends on the logic levels present at the inputs. AND Gate OR Gate Exclusive-OR Gate NAND Gate NOR Gate Exclusive-NOR Gate The OR Operation & Gate Truth Table & Boolean Expression 4-input OR Gate Symbol or Schematic The output of an OR gate is HIGH whenever one or more inputs are HIGH 4-Input OR Gate

4 Summary of OR operation Example of the use of an OR gate in an Alarm system Produce a result of 1 whenever any input is 1. Otherwise 0. An OR gate is a logic circuit that performs an OR operation on the circuit's input The expression x=a+b is read as x equals A OR B Example 2 Example

5 Review Questions The AND Operation & AND Gate What is the only set of input conditions that will produce a LOW output for any OR gate? all inputs LOW Write the for a six-input OR gate X=A+B+C+D+E+F If the A input in previous example is permanently kept at the 1 level, what will the resultant output waveform be? constant HIGH The output of an AND gate is HIGH only when all inputs are HIGH Input AND Gate Summary of the AND operation Review Questions The AND operation is performed the same as ordinary multiplication of 1s and 0s. An AND gate is a logic circuit that performs the AND operation on the circuit s inputs. An AND gate output will be 1 only for the case when all inputs are 1; for all other cases the output will be 0. The expression x=a B is read as x equals A AND B. What is the only input combination that will produce a HIGH at the output of a five-input AND gate? all 5 inputs = 1 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? A LOW input will keep the output LOW True or false: An AND gate output will always differ from an OR gate output for the same input conditions. False

6 The NOT Operation & Inverter Boolean Operations and Expressions Sum (OR) = = = = 1 Product (AND) 0 0 = = = = 1 The output of an inverter is always the complement (opposite) of the input The NAND Gate The NOR Gate The output of a NAND gate is HIGH whenever one or more inputs are LOW. The output of a NOR gate is LOW whenever one or more inputs are HIGH

7 Describing logic circuits algebraically 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 Parentheses 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. Therefore the two cases of interpretations are : Circuits Contain INVERTERs More Examples 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

8 Precedence 1. First, perform all inversions of single terms 2. Perform all operations with parentheses 3. Perform an AND operation before an OR operation unless parentheses indicate otherwise 4. 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 More Example Implementing Circuits From Boolean Expressions When the operation of a circuit is defined by a Boolean expression, we can draw a logic-circuit diagram directly from that expression

9 Example Draw the circuit diagram to implement the expression x = ( A + B )( B + C ) Review Question Draw the circuit diagram that implements the expression x = ABC( A + D) using gates having no more than three inputs

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