1.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Digital Electronics and Logic Design
Logic Gates
2.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Logic Gates
• A logic gate is an electronic circuit/device which makes the logical
decisions. To arrive at this decisions, the most common logic gates used
are OR, AND, NOT, NAND, and NOR gates. The NAND and NOR gates are
called universal gates. The exclusive-OR gate is another logic gate which
can be constructed using AND, OR and NOT gate.
• Logic gates have one or more inputs and only one output. The output is
active only for certain input combinations. Logic gates are the building
blocks of any digital circuit. Logic gates are also called switches.
2
3.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
3
Basic logic gates
• Not
• And
• Or
• Nand
• Nor
• Xor
x
x
x
y
xy x
y
xyz
z
x yx
y
x
y
x+y+z
z
x
y
xy
x yx
y
x yx
y
4.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Digital Electronics and Logic Design
Logic Gates
5.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
AND Function
Output Y is TRUE if inputs A AND B are TRUE,
else it is FALSE.
Logic Symbol
Text Description
Truth Table
Boolean Expression
AND
A
B
Y
INPUTS OUTPUT
A B Y
0 0 0
0 1 0
1 0 0
1 1 1
AND Gate Truth Table
Y = A x B = A • B = AB
AND Symbol
6.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
OR Function
Output Y is TRUE if input A OR B is TRUE, else it
is FALSE.
Logic Symbol
Text Description
Truth Table
Boolean Expression Y = A + B
OR Symbol
A
B
YOR
INPUTS OUTPUT
A B Y
0 0 0
0 1 1
1 0 1
1 1 1
OR Gate Truth Table
7.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
NOT Function (inverter)Output Y is TRUE if input A is FALSE, else it is
FALSE. Y is the inverse of A.
Logic Symbol
Text Description
Truth Table
Boolean Expression
INPUT O UTPUT
A Y
0 1
1 0
NOT Gate Truth Table
A YNOT
NOT
Bar
Y = A
Y = A’
Alternative Notation
Y = !A
8.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
NAND FunctionOutput Y is FALSE if inputs A AND B are TRUE,
else it is TRUE.
Logic Symbol
Text Description
Truth Table
Boolean Expression
A
B
YNAND
A bubble is an inverter
This is an AND Gate with an inverted output
Y = A x B = AB
INPUTS OUTPUT
A B Y
0 0 1
0 1 1
1 0 1
1 1 0
NAND Gate Truth Table
9.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
NOR FunctionOutput Y is FALSE if input A OR B is TRUE, else it
is TRUE.
Logic Symbol
Text Description
Truth Table
Boolean Expression Y = A + B
A
B
YNOR
A bubble is an inverter.
This is an OR Gate with its output inverted.
INPUTS OUTPUT
A B Y
0 0 1
0 1 0
1 0 0
1 1 0
NOR Gate Truth Table
10.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
10
11.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Circuit-to-Truth Table Example
OR
A
Y
NOT
AND
B
C
AND
2# of Inputs = # of Combinations
2 3 = 8
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
A B C Y
12.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Circuit-to-Truth Table Example
OR
A
Y
NOT
AND
B
C
AND
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
A B C Y
0
0
0
0
1
0
0
0
13.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Circuit-to-Truth Table Example
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
A B C Y
0
OR
A
Y
NOT
AND
B
C
AND
0
0
1
0
1
1
1
1
14.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Circuit-to-Truth Table Example
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
A B C Y
0
1
0
OR
A
Y
NOT
AND
B
C
AND
0
1
0
0
1
0
0
0
15.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Circuit-to-Truth Table Example
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
A B C Y
0
1
0
0
OR
A
Y
NOT
AND
B
C
AND
0
1
1
0
1
1
1
1
16.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Circuit-to-Truth Table Example
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
A B C Y
0
1
0
1
0
OR
A
Y
NOT
AND
B
C
AND
1
0
0
0
0
0
0
0
17.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Circuit-to-Truth Table Example
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
A B C Y
0
1
0
1
0
0
OR
A
Y
NOT
AND
B
C
AND
1
0
1
0
0
0
0
0
18.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Circuit-to-Truth Table Example
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
A B C Y
0
1
0
1
0
0
0
OR
A
Y
NOT
AND
B
C
AND
1
1
0
1
0
0
1
1
19.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Circuit-to-Truth Table Example
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
A B C Y
0
1
0
1
0
0
1
0
OR
A
Y
NOT
AND
B
C
AND
1
1
1
1
0
0
1
1
20.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
NAND Gate – Special Application
INPUTS OUTPUT
A B Y
0 0 1
0 1 1
1 0 1
1 1 0
A
B
YNAND
TNANDS
S T
00
1
0 1
1 0
Equivalent To An Inverter Gate
21.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
NOR Gate - Special Application
S T
00
1
0 1
1 0
Equivalent To An Inverter Gate
TS NOR
A
B
YNOR
INPUTS OUTPUT
A B Y
0 0 1
0 1 0
1 0 0
1 1 0
22.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Boolean logic
22
23.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 23
Chapter 3 Objectives
• Understand the relationship between Boolean logic and
digital computer circuits.
• Learn how to design simple logic circuits.
• Understand how digital circuits work together to form
complex computer systems.
24.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 24
3.2 Boolean Algebra
25.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Boolean Functions to Logic Circuits
• Any Boolean expression can be converted to a logic
circuit made up of AND, OR and NOT gates.
step 1: add parentheses to expression to fully
define order of operations - A+(B C ))
step 2: create gate for “last” operation in expression
gate’s output is value of expression
gate’s inputs are expressions combined by operation
A
A+B C
(B C ))
step 3: repeat for sub-expressions and continue until done
Number of simple gates needed to implement expression equals number of operations in
expression.
– so, simpler equivalent expression yields less expensive circuit
– Boolean algebra provides rules for simplifying expressions
26.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Basic Identities of Boolean Algebra
1. X 0 X
3.X 1 1
5.X X X
7.X X ’ 1
9.(X ’)’ X
10.X Y Y X
12.X Y Z) (X Y) Z
14.X Y Z) X Y X Z
16. X Y) X Y
2. X 1 X
4. X 0 0
6. X X X
8. X X ’ 0
11. X Y Y X
13. X Y Z ) (X Y ) Z
15. X Y Z ) X Y ) X Z )
17. X Y)’ = X Y
commutative
associative
distributive
DeMorgan’s
Identities define intrinsic properties of Boolean algebr
Useful in simplifying Boolean expressions
27.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Verifying Identities Using Truth Tables
• Can verify any logical equation with small number
of variables using truth tables.
• Break large expressions into parts, as needed.
X Y Z ) X Y ) X Z )
Y Z
0
0
0
1
0
0
0
1
XYZ
000
001
010
011
100
101
110
111
X Y Z )
0
0
0
1
1
1
1
1
X Y
0
0
1
1
1
1
1
1
X Z
0
1
0
1
1
1
1
1
X Y ) X Z )
0
0
0
1
1
1
1
1
X Y ) X Y
XY
00
01
10
11
X Y
1
0
0
0
X Y )
1
0
0
0
28.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
DeMorgan’s Law
29.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
DeMorgan’s Laws for n Variables
• We can extend DeMorgan’s laws to 3 variables by
applying the laws for two variables.
(X Y Z) X Y Z)) - by associative law
X Y Z) - by DeMorgan’s law
X Y Z ) - by DeMorgan’s law
X Y Z - by associative law
(X Y Z) X Y Z)) - by associative law
X Y Z) - by DeMorgan’s law
X Y Z ) - by DeMorgan’s law
X Y Z - by associative law
• Generalization to n variables.
30.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Simplification of Boolean Expressions
F=X YZ +X YZ +XZ
Y
Z
X
Y
Z
X
Y
Z
X
F=X Y(Z +Z )+XZ
by identity 14
F=X Y 1+XZ
=X Y +XZ by identity 2
by identity 7
31.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
The Duality Principle
• The dual of a Boolean expression is obtained by
interchanging all ANDs and ORs, and all 0s and 1s.
– example: the dual of A+(B C )+0 is A (B+C ) 1
• The duality principle states that if E1 and E2 are
Boolean expressions then
E1= E2 dual (E1)=dual(E2)
where dual(E) is the dual of E. For example,
A+(B C )+0 = (B C )+D A (B+C ) 1 = (B +C ) D
Consequently, the pairs of identities (1,2), (3,4),
(5,6), (7,8), (10,11), (12,13), (14,15) and (16,17)
all follow from each other through the duality
32.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
32
33.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Sum of Products Form
• Minterm: If a product term (AND) of a function of ‘n’
variables, contain all the variables in complemented form
or uncomplemented form, then it is called a “minterm” or
“standard product”.
Sum of product method:
• The four possible ways to AND two input signals that are in
complemented and
• uncomplemented from. These outputs are called
fundamental products. The following table lists
• each fundamental product next to the input conditions
producing a high input for instance, AB is
• high when A and B are low. AB is high when A is high and so
34.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Product of Sums Form
• The product of sums is the second standard form for
Boolean expressions.
product-of-sums-expression = s-term s-term ... s-term
s-term = literal literal literal
Example. (X +Y +Z )(X +Z)(X+Y)(X+Y+Z)
• A maxterm is a sum term that contains every variable, in
complemented or uncomplemented form.
Example. in exp. above, X +Y +Z is a maxterm, but X +Z is not
• A product of maxterms expression is a product of sums
expression in which every term is a maxterm
Example. (X +Y +Z )(X +Y+Z)(X+Y+Z )(X+Y+Z) is product of maxterms
expression that is equivalent to expression above
35.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
take the whole compliment, since POS is a dual to the
SOP form,
In the following we apply de-morgan's laws which
are : (a+b)' = a' b' ; (ab)' = a' + b'
(A’CD + E’F + BCD)'
= (A'CD + E'F)' * (BCD)'
= (A'CD)' * (E'F)' * (BCD)'
= ((A')' + C' + D' ) * ( (E')' + F' ) * (B' + C' + D')
= (A + C' + D' ) * ( E + F') * (B' + C' + D')
which is the product of sums, that is POS form!
36.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
NAND and NOR Gates(Universal Gates)
• In certain technologies (including CMOS), a NAND
(NOR) gate is simpler & faster than an AND (OR)
gate.
• Consequently circuits are often constructed using
NANDs and NORs directly, instead of ANDs and ORs.
• Alternative gate representations makes this easier.
X
Y (X Y)NAND Gate (X Y)
X
Y
NOR Gate
= =
==
37.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Exclusive Or and Odd Function
• The odd function on n variables is 1 when an odd
number of its variables are 1.
– odd(X,Y,Z) = XY Z + X YZ + X Y Z + XYZ = X Y Z
– similarly for 4 or more variables
• Parity checking circuits use the odd function to
provide a simple integrity check to verify
correctness of data.
EXOR gate
Alternative Implementation
A
B
The EXOR function is defined by A B = AB + A B.
A
AB
+A B
B
38.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 38
3.3 Logic Gates
• NAND and NOR are known as
universal gates because they are
inexpensive to manufacture and any
Boolean function can be constructed
using only NAND or only NOR gates.
• Gates can have multiple inputs and more
than one output.
A second output can be provided for
the complement of the operation.
We’ll see more of this later.
39.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 39
3.4 Digital Components
• Combinations of gates implement Boolean functions.
• The circuit below implements the function:
• This is an example of a combinational logic circuit.
• Combinational logic circuits produce a specified output
(almost) at the instant when input values are applied.
Later we’ll explore circuits where this is not the case.
40.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 40
3.5 Combinational Circuits
• As we see, the sum can be
found using the XOR operation
and the carry using the AND
operation.
• Combinational logic circuits
give us many useful devices.
• One of the simplest is the half
adder, which finds the sum of
two bits.
• We can gain some insight as to
the construction of a half adder
by looking at its truth table,
shown at the right.
41.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 41
3.5 Combinational
Circuits
• We can change our half adder into
to a full adder by including gates for
processing the carry bit.
• The truth table for a full adder is
shown at the right.
FULL ADDER
HALF ADDER
42.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 42
3.5 Combinational Circuits
• Just as we combined half adders to make a full adder,
full adders can connected in series.
• The carry bit “ripples” from one adder to the next;
hence, this configuration is called a ripple-carry adder.
74LS283
This is a 4-bit adder that
you can program as part
of your Project.
43.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 43
3.5 Combinational Circuits
• Decoders are another important type of combinational circuit.
• Among other things, they are useful in selecting a memory location based on
a binary value placed on the address lines of a memory bus.
• Address decoders with n inputs can select any of 2n
locations.
• This is what a 2-to-4 decoder looks like on the inside.
If x = 0 and y = 1,
which output line
is enabled?
44.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 44
3.6 Sequential Circuits
• Another modification of the SR flip-flop is the D flip-flop, shown below with its
characteristic table.
• The output of the flip-flop remains the same during subsequent clock pulses.
The output changes only when the value of D changes.
• The D flip-flop is the fundamental circuit of
computer memory.
D flip-flops are usually illustrated using the
block diagram shown here.
The previous state doesn’t matter. Totally dependent on
state of D
45.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 45
Appendix - 3.2 Boolean Algebra
• Boolean algebra is a mathematical system for the
manipulation of variables that can have one of two values.
– In formal logic, these values are “true” and “false.”
– In digital systems, these values are “on” and “off,” 1 and 0, or
“high” and “low.”
• Boolean expressions are created by performing operations
on Boolean variables.
– Common Boolean operators include AND, OR, and NOT.
46.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 46
3.2 Boolean Algebra
• A Boolean operator can be
completely described using a truth
table.
• The truth table for the Boolean
operators AND and OR are shown at
the right.
• The AND operator is also known as a
Boolean product. The OR operator
is the Boolean sum.
47.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 47
3.2 Boolean Algebra
• The truth table for the
Boolean NOT operator is
shown at the right.
• The NOT operation is most
often designated by an
overbar. It is sometimes
indicated by a prime mark ( ‘
) or an “elbow” ( ).
48.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 48
3.2 Boolean Algebra
• A Boolean function has:
• At least one Boolean variable,
• At least one Boolean operator, and
• At least one input from the set {0,1}.
• It produces an output that is also a member of the
set {0,1}.
Now you know why the binary numbering
system is so handy in digital systems.
49.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 49
3.2 Boolean Algebra
• The truth table for the
Boolean function:
is shown at the right.
• To make evaluation of the
Boolean function easier, the
truth table contains extra
(shaded) columns to hold
evaluations of subparts of
the function.
50.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 50
3.2 Boolean Algebra
• As with common
arithmetic, Boolean
operations have rules of
precedence.
• The NOT operator has
highest priority, followed
by AND and then OR.
• This is how we chose the
(shaded) function subparts
in our table.
51.
Subject Name Code Credit Hours
Digital Electronics and Logic Design DEL-244 3
Chapter 3: Digital Logic 51
• Computers are implementations of Boolean logic.
• Boolean functions are completely described by truth
tables.
• Logic gates are small circuits that implement Boolean
operators.
• The basic gates are AND, OR, and NOT.
– The XOR gate is very useful in parity checkers and
adders.
• The “universal gates” are NOR, and NAND.
Chapter 3 Conclusion
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