Department of Computer and Information Science, School of Science, IUPUI CSCI 240 Digital Logic.
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Transcript of Department of Computer and Information Science, School of Science, IUPUI CSCI 240 Digital Logic.
Department of Computer and Information Science,School of Science, IUPUI
CSCI 240
Digital Logic
Dale Roberts
Boolean Algebra to Logic GatesBoolean Algebra to Logic Gates
Logic circuits are built from components called Logic circuits are built from components called logic gates.logic gates.The logic gates correspond to Boolean operations The logic gates correspond to Boolean operations +, *, +, *, ’.’.
Binary operations have two inputs, unary has oneBinary operations have two inputs, unary has one
OR+
AND*
NOT’’
Dale Roberts
ANDAND
A
B
A*B
Logic Gate:
Series Circuit:
A B
AA BB A*BA*B
00 00 00
00 11 00
11 00 00
11 11 11
Truth Table:A*B
Dale Roberts
A
B
A+B
Logic Gate:
Parallel Circuit:
A
B
AA BB A+BA+B
00 00 00
00 11 11
11 00 11
11 11 11
Truth Table:A+B
OROR
Dale Roberts
NOTNOT
A
A’ or A
Logic Gate:(also called an inverter)
Single-throwDouble-poleSwitch:
A
a A
0 1
1 0
Truth Table:A’ or A
Dale Roberts
nn-input Gates-input Gates
Because + and * are binary operations, they can Because + and * are binary operations, they can be cascaded together to OR or AND multiple be cascaded together to OR or AND multiple inputs.inputs.
AB
C
ABC
A+B+C
A+B+C
AB
ABC
ABC
ABC
Dale Roberts
nn-bit Inputs-bit Inputs
For convenience, it is sometimes useful to think For convenience, it is sometimes useful to think of the logic gates processing of the logic gates processing nn-bits at a time. -bits at a time.
This really refers to This really refers to nn instances of the logic instances of the logic
gate, not a single logic date with gate, not a single logic date with nn-inputs.-inputs.
1101100101
01001101111101110111
10001111
0011110000001100
110001 001110
Dale Roberts
Logic Circuits Logic Circuits ≡≡ Boolean Expressions Boolean Expressions
All logic circuits are equivalent to Boolean expressions and All logic circuits are equivalent to Boolean expressions and any boolean any boolean expression can be rendered as a logic circuit.expression can be rendered as a logic circuit.AND-OR logic circuits are equivalent to sum-of-products form.AND-OR logic circuits are equivalent to sum-of-products form.Consider the following circuits:Consider the following circuits:
A
CB abc
aBc
Ab
y=abc+aBc+Ab
y
A
B
C
Y
y=aB+Bc
Dale Roberts
NAND and NOR GatesNAND and NOR Gates
NAND and NOR gates can greatly simplify circuit NAND and NOR gates can greatly simplify circuit diagrams. As we will see, can you use these gates diagrams. As we will see, can you use these gates wherever you could use AND, OR, and NOT.wherever you could use AND, OR, and NOT.
NAND
NOR
AA BB AABB
00 00 11
00 11 11
11 00 11
11 11 00
AA BB AABB
00 00 11
00 11 00
11 00 00
11 11 00
Dale Roberts
XOR and XNOR GatesXOR and XNOR Gates
XOR is used to choose between two mutually XOR is used to choose between two mutually exclusive inputs. Unlike OR, XOR is true only exclusive inputs. Unlike OR, XOR is true only when one input or the other is true, not both.when one input or the other is true, not both.
XOR
XNOR
AA BB AABB
00 00 00
00 11 11
11 00 11
11 11 00
A B A B
0 0 1
0 1 0
1 0 0
1 1 1
Dale Roberts
Properties of NAND AND NORProperties of NAND AND NOR
NAND and NOR have special properties, but NAND and NOR have special properties, but neither satisfies the distributive or associative neither satisfies the distributive or associative laws.laws.
NANDNAND NORNOR
xx1=X1=X xx0=X0=X
x x 0=1 0=1 x x 1=0 1=0
x x x=X x=X x x x=X x=X
x x y=X+Y y=X+Y x x y=XY y=XY
X X Y=x+y Y=x+y X X Y=xy Y=xy
not (x not (x y)=xy y)=xy not (x not (x y)=x+y y)=x+y
It should be clear by looking at these properties that It should be clear by looking at these properties that NAND and NOR are duals.NAND and NOR are duals.
Dale Roberts
NAND and NOR as Universal Logic GatesNAND and NOR as Universal Logic Gates
Any logic circuit Any logic circuit can be built using can be built using only NAND gates, only NAND gates, or only NOR or only NOR gates. They are gates. They are the only logic the only logic gate needed.gate needed.
Here are the Here are the NAND NAND equivalents:equivalents:
Dale Roberts
NAND and NOR as Universal Logic Gates (cont)NAND and NOR as Universal Logic Gates (cont)
Here are the NOR Here are the NOR equivalents:equivalents:
NAND and NOR NAND and NOR can be used to can be used to reduce the reduce the number of number of required gates in required gates in a circuit.a circuit.
Dale Roberts
Example ProblemExample Problem
A hall light is controlled by two light switches, A hall light is controlled by two light switches, one at each end. Find (a) a truth function, (b) a one at each end. Find (a) a truth function, (b) a Boolean expression, and (c) a logic network that Boolean expression, and (c) a logic network that allows the light to be switched on or off by allows the light to be switched on or off by either switch.either switch.
xx yy f(f(xx,,yy))
00 00 00
00 11 11
11 00 11
11 11 00
(What kind of gate has this truth table?
Let Let xx and and yy be the switches: be the switches:
Dale Roberts
Example (cont)Example (cont)
One possible equation is the complete sum-of-products form:One possible equation is the complete sum-of-products form:
f(x,y) = xY + Xyf(x,y) = xY + Xy
Use The Most Complex Machine Use The Most Complex Machine
xLogicCircuit Module to implement thexLogicCircuit Module to implement the
equation.equation.
xx yy f(f(xx,,yy))
00 00 00
00 11 11
11 00 11
11 11 00
Dale Roberts
AcknowledgementsAcknowledgements
Eck, David. Eck, David. The Most Complex MachineThe Most Complex Machine
Gersting, Judith, Gersting, Judith, Mathematical Structures for Mathematical Structures for Computer ScienceComputer Science