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Lecture 5: Logic Gates

January 9, 2007 •Basic gates•Boolean Algebra•Boolean’s Theorem

Binary Variables and Functions „^ Basic

logic

operations:

^ AND ^ OR ^ NOT inputs^ „ Gate—A circuit that performs a specified logic operation

output

inputs

output

input

output

AND GateTiming Diagrams

Timing Diagrams

The OR Gate „ Two or more inputs and one output „ Symbol X Y „ Logic expression^

^ Symbol is + , Not addition^ ^ Z = X + Y „ Logic operation^ ^ Output Z is 1 when at least one of the inputs X or Y is 1^ ^ Output Z is 0 when all inputs X, Y are 0

Z^

≥^1

X Y

Z

OR Truth Table

INVERTER „ Unary „ Symbol „ Logic Operation

^ Z = X ^ Z = X’ „ Inverter Operation ^ “The output is opposite the input”

x^ indicate inversion or complementation

z^

x^

z

INVERTER

INVERTER Truth Table

t^1

t^2 InputOutput

„^ Timing diagrams „^ Truth table

More Inputs „^ Work same way „^ What’s output?

What’s output?1 if all inputsare 1 What’s output?1 if at least one input is 1

The NAND Gate „^ A universal gate^

^

Used in combination to perform AND, OR and inverter operations. „^ NAND is a contraction of NOT-AND^ ^

AND function with complemented output „^ Symbols „^ Logic Expression^ ^

Z=XY „^ Logic Operation^ ^ X Y Output Z is 0 when all inputs X, Y are 1^ ^ Output Z is 1 when at least one of the inputs , X or Y is 0

Z^

X Y

Z

Digital Electronics

NAND Gate is a universal gate X

Z^

x^

z

X Y

Z

X Y

Z

X Y^

Z

X Y

Z

NOR Gate is a universal gate

x^

z X Y^

Z

X Y

Z

x

z

X Y

Z

X Y^

Z

The NOR Gate „^ Timing Diagrams „^ Truth Table

X^ Y

Z

Z^

The Exclusive-OR and Exclusive-NOR Gates „^ Formed by a combination of other gates „^ Important in many application „^ Symbol of Exclusive-OR (XOR)^ X Y

Z

X^ Y^

Z

XOR truth table

• Logic Operation•Output Z is 1 when X and Y are different.•Output Z is 0 when X and Y are same.

X Y

Z

Exercise 1: Least Challenging Complete the sentences:^ ◊

If all inputs of an OR gate are low, the outputis_________ ◊ If any input of an AND gate is low, the output is_________ ◊ The output of a NOT gate is always the___________ of the input.

Boolean Algebra „^ Mathematics of digital system „^ Important in the analysis of logic circuit „^ Binary variable– a symbol (letter) used to represent alogical quantity. „^ Example binary variables and their values:^

^ A = 0^ ^ B = 1 „^ Complement—the inverse of a variable^ ^ The complement of A is A or A’.