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![Page 1: Boole Algebra and Logic Series - · PDF fileS1 Teknik Telekomunikasi Fakultas Teknik Elektro 2016/2017 Boole Algebra and Logic Series CLO1-Week2-Basic Logic Operation and Logic Gate](https://reader031.vdocuments.mx/reader031/viewer/2022022501/5aa986b57f8b9a7c188cf528/html5/thumbnails/1.jpg)
S1 Teknik Telekomunikasi
Fakultas Teknik Elektro
2016/2017
Boole Algebra and Logic Series
CLO1-Week2-Basic Logic
Operation and Logic Gate
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• Understand the basic theory of Boolean
• Understand the basic algebra law in Boolean
based on set theory
• Understand how to operate algebra law by using
basic logic operation
• Knowing about Basic Logic Gate to express Basic
Operation
2
Outline
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Boolean Algebra
• Boolean algebra provides the operations and the rules for working with the set B = {0, 1}
Why only “0” and “1”?
• 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.”
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Operation – “Union” / “OR”
Set Operation - Union
“A union B” is the set of all elements that are in A, or B, or both
Boolean Operation - OR
• The OR operator is the
Boolean sum
Operator: “+”
Logic Gate:
4
A B
A
B
Z
A
S
BA + B
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Operation – “Intersection” / “AND”
Set Operation - Intersect
“A intersect B” is the set of all elements that are in both A and B.
Boolean Operation - AND
• The AND operator is also
known as a Boolean
product
Operator: “.” Logic Gate:
5
A B
A
B
Z
A
S
B
A . B
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Operation – “Complement” / “NOT”
Set Operation - Complement
“A complement,” or “not A” is the set of all elements not in A
Boolean Operation - NOT
• The NOT operation is
most often designated by
an overbar. It’s also
called inverter
Operator: “ ‘ “ or “ “ Operator gate:
6
A
A Z
A
A
S
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Operation – “Comp. Of Union” / “NOR”
Set Operation
Is the complement of “A union B”
Boolean Operator - NOR
• The NOR operation is
combination of NOT and
OR operation
Operator :
Logic Gate:
7
A
S
BA + B
(A B)’
A
B
Z
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Operation – “Comp. Of Intersect” / “NAND”
Set Operation
Is the complement of “A intersect B”
Boolean Operation - NAND
• The NAND operation is
combination of NOT and
AND operation
Operator :
Logic Gate:
8
A
S
B
A . B
(A B)’
A
B
Z
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Operation – Symmetric Differece / “XOR”
Set Operation – Sym. Diff.
Boolean Operation - XOR
• The output of the XOR
operation is true only
when the values of the
inputs differ
Operator: “ ⊕ “
Logic Gate:
9
(A B) - (A B)
A
S
BB
A
B
Z
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• 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}.
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Now you know why the binary numbering system is
so handy in digital systems
Digital System is based on PULSE SIGNAL, which
valued “0” or “1”
Boolean Function
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Combination in Boolean Function
Multiple Inputs? (>2 inputs)
Multiple Gate/Operator?
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Multiple Inputs-Outputs?
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• The three simplest gates are the AND, OR, and NOT
gates.
• They correspond directly to their respective Boolean
operations, as you can see by their truth tables.
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Let’s start from the simplest
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3.3 Logic Gates
• Another very useful gate is the exclusive OR
(XOR) gate.
• The output of the XOR operation is true only when
the values of the inputs differ.
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Note the special symbol
for the XOR operation.
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3.3 Logic Gates
• NAND and NOR
are two very
important gates.
Their symbols and
truth tables are
shown at the right.
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AND and OR gate with 3 inputs
A B C A+B+C
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1
A B C A.B.C
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
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Example of Simple Series
A
B
?
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Truth table of the Series
A B A B A • B A • B
0 0 1 1 1 0
0 1 1 0 0 1
1 0 0 1 0 1
1 1 0 0 0 1
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Example in Implementation
BABAf
A B f
0 0 0 0 0
0 1 0 1 1
1 0 1 0 1
1 1 0 0 0
BA BA
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Example in Implementation
BABAf A
B
f
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Chips/ IC Digital Dasar
• To implement the logic diagram, we use the digital electronic series of logic IC/chips
• The kind of Logic Chip there are in market is IC TTL (Transistor-transistor Logic) or MOS
• Those Chip are identified by part number or model number.
• IC type of standard digital series is started by number 74, 4, or 14.
– 7404 is an inverter
– 7408 is an AND
– 7432 is an OR
– 4011B is a NAND
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Chips
• Basic Logic Chip is in DIP form (dual in package) with even pins. The usual form has 14-pins
• Pin 1 marked by dot or half-circle
• The next pin is read by CCW way
Pin 1 Pin 7
Pin 14 Pin 8
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Chips • Chips need voltage to be
operated
• Vcc is used to interface of 5 volts and VCC pin usually placed at last number of pins (for DIP14 so, VCC is at pin-14)
• Ground Pin usually placed at last pin at same side of first pin (for DIP14, so GND is at no.7)
Voltage
Ground
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Example of Basic Logic IC TTL
• 74LS00 : Quad 2 input NAND Gate
14 13 12 11 10 9 8
1 2 3 4 5 6 7
VCC
GND
14 13 12 11 10 9 8
1 2 3 4 5 6 7
VCC
GND
74LS08 : Quad 2 input AND Gate
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Example of Basic Logic IC TTL
• 74LS02 : Quad 2 input NOR Gate
14 13 12 11 10 9 8
1 2 3 4 5 6 7
VCC
GND
74LS32 : Quad 2 input OR Gate
891011121314
7654321
VCC
GND
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Example of Basic Logic IC TTL
• 74LS04 : Hex Inverter
14 13 12 11 10 9 8
1 2 3 4 5 6 7
VCC
GND
74LS86 : Quad 2 input XOR Gate
14 13 12 11 10 9 8
1 2 3 4 5 6 7
VCC
GND
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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.
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3.3 Logic 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.
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3.4 Digital Components
• The main thing to remember is that combinations
of gates implement Boolean functions.
• The circuit below implements the Boolean
function:
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We simplify our Boolean expressions so
that we can create simpler circuits.
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3.5 Combinational Circuits
• 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.
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3.5 Combinational Circuits
• As we see, the sum can be
found using the XOR
operation and the carry
using the AND operation.
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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.
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3.5 Combinational Circuits
• How can we change the
half adder shown below
to make it a full adder?
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3.5 Combinational Circuits
• Here’s our completed full adder.
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