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CHAPTER 2:
INTRODUCTION TO
LOGIC CIRCUITS
SPRING 2026
Soontae Kim, School of Computing., KAIST
Example of Binary logic
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Binary switch
“closed”
“open”
x = 0
Input
variable
x = 1
(a) Two states of a switch
S
x
(b) Symbol for a switch
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Application of a switch
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S
Battery
x
Light
(a) Simple connection to a battery
Output
State of the light : L =1 if the light is on, L=0 (off)
The state of the light can be described as a function of the input
variable x
L(x) = x : logic function
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Two basic functions
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Power
supply
S
S
x1
x2
Light
L(x1,x2) = x1 • x2 or (x1 x x2 )
where L=1 if x1 = 1 and x2=1,
L=0 otherwise
(a) The logical AND function (series connection)
S
x1
Power
supply
S
Light
L(x1,x2) = x1 + x2
where L=0 if x1 = x2=0,
L=1 otherwise
x2
(b) The logical OR function (parallel connection)
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Series-Parallel Connection
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S
X1
Power
supply
S
S
X3
Light
X2
Logic function ?
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Inversion
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R
Power
supply
x
S
Light
Inverse of a value
Complement of a value
The NOT operation
Several commonly used notations
x = x’ = !x = NOT x = ~x
The complement operation can be applied to more complex
operations
f(x1,x2) = x1+x2
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Truth Tables: AND and OR
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Useful aid for depicting information involving logic
functions
Grow exponentially in size with the number of
variables
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Three input AND and OR
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Table size : n inputs ? = 2n
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Possible logic functions of two
variables
How many possible functions?
16 possible functions of 2 input variables
X
Y
2𝑛
n inputs : 2
F
X Y
0
0
0
0
0 0
0 1
1 0
1 1
0
0
0
1
0
0
1
0
0
0
0
1
1
X
0
1
0
0
0
1
0
1
Y
X and Y
0
1
1
0
0
1
1
1
1
0
0
0
1
0
0
1
X or Y
X xor Y
1
0
1
0
1
0
1
1
X=Y
X nor Y
1
1
0
0
1
1
0
1
1
1
1
0
not X
not Y
1
1
1
1
1
X nand Y
Any boolean function can be generated by a combination of functions?
Complete Set : a set of function is complete iff every Boolean function can be generated by
a combination of the functions
{NAND}, {NOR}, {AND, NOT}
{AND, OR} ?
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Basic Gates
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What is a logic gate or simply a gate?
A circuit element implementing electronically
a certain logic function with transistors
x1
x2
x1 x2
x1
x2
x1 x2 xn
xn
(a) AND gates
x1
x2
x1 + x2
x1
x2
xn
x1 + x2 + + xn
x
x
(c) NOT gate
(b) OR gates
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An AND-OR Function
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S
Power
supply
X1
S
S
X3
Light
X2
A network of gates: logic network or logic circuit
x
1
x
2
x
3
x
x
x
f = (x + x ) x
1
2
3
1
2
3
1
f = ( x + x ) x3 1
1
2
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Analysis of a Logic Network
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Analysis and Synthesis
x1
x2
0 → 0 →1 → 1
1 →1 →0 →0
A
0 →0 →0 →1
0 → 1 →0 → 1
x
f (x , x )
0
0
1
1
0
1
0
1
1
1
0
1
1
2
1
2
f
B
f = x1 + x1 ×x2
(a) Network that implements
x
1 →1 →0 →1
A
B
1
0
1
0
0
0
0
1
analysis
circuit
function
synthesis
(b) Truth table
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Timing Diagram
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Timing Diagram
Changes in signals at various points are presented in graphical form
Also useful method for indicating the functional behavior
x1
x2
0 → 0 →1 → 1
1 →1 →0 →0
A
0 → 1 →0 → 1
0 →0 →0 →1
1 →1 →0 →1
f
B
x1 1
0
x2 1
0
1
A 0
1
B 0
f 1
0
waveform
Time
(c) Timing diagram
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Functionally Equivalent Networks
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x1 + x1 • x2 = x1 + x2
x1
x2
0 → 0 →1 → 1
1 →1 →0 →0
A
0 → 1 →0 → 1
0 →0 →0 →1
(a) Network that implements
x
x
0 →0 →1 →1
1 →1 →0 →1
f
B
f = x1 + x1 ×x2
1 →1 →0 →0
1
1 →1 →0 → 1
0 →1 →0 →1
g
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(d) Network that implements
g = x +x
1 2
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Functionally Equivalent
Networks(cont’d)
A logic function can be implemented
with a variety of different networks, probably having different
costs
This raises an important question
How do we find the best implementation for a given function?
◼ What is the best implementation?
Many techniques for synthesizing logic functions
◼ Discussed in Chapter 4
◼ Through the Truth Table
◼ Algebraic(Boolean) manipulation of logic expressions
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What will we learn?
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Logic functions and circuits
Boolean Algebra
Logic gates and Synthesis
CAD tools and VHDL
Read Section 2.9 and 2.10
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Boolean Algebra
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In 1849 George Boole published
In the late 1930 Claude Shannon showed
A scheme for the algebraic description of processes involved in logical thought and
reasoning
Boolean algebra provides an effective means of describing circuits built with switches
A Boolean algebraic structure consists of
Elements taking 0 or 1
Binary operation { +(OR), • (AND) }
And a unary operation { ‘ (NOT)}
The following axioms hold
◼ 0∙0 = 0
◼ 1+1=1
◼ 1 ∙1=1
◼ 0+0=0
◼ 0∙1=1∙0=0
◼ 1+0=0+1=1
◼ If x=0, then ‘x=1
◼ If x=1, then ‘x=0
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Theorems of Boolean Algebra
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Single-variable theorems
x ∙0=0 x+1=1
x ∙1=x x+0=x
x ∙x=x x+x=x
x ∙’x=0 x+’x=1
‘’x=x
Duality
A dual of a Boolean expression is derived by replacing • by +, + by • , 0 by 1,
and 1 by 0, and leaving variables unchanged
Dual of any true statement is also true
X+Y+ …X•Y•…
DeMorgan’s theorem
X•Y=X+Y
X+Y= X•Y
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Two- and three- variable properties
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Proof of DeMorgan’s Theorem
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Notation and Precedence of Operations
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Logical sum and logical product operations
The OR and AND operations
Say simply “sum” and “product”
Product terms and Sum terms
◼ x•y : product term (xy)
◼ x+y : sum term
Precedence
In the order: NOT, AND, and then OR
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