S1 Teknik Telekomunikasi
Fakultas Teknik Elektro
2016/2017
Boole Algebra and Logic Series
CLO1-Week2-Basic Logic
Operation and Logic Gate
• 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
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Outline
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
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
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
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
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
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
• 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
Combination in Boolean Function
Multiple Inputs? (>2 inputs)
Multiple Gate/Operator?
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Multiple Inputs-Outputs?
• 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
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.
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
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.
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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