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Claude Shannon
•  Father of information
theory
Logic Gates
CS 231
Dianna Xu
•  His master thesis was
the foundation of
digital circuit design
theory.
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Circuits
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Boolean algebra
•  Boolean – a variable that is either true or
false
•  Boolean algebra – logical calculus of truth
values
•  Single switch
•  Switches in series
–  Very similar to Boolean logic
–  Variables can only be 1 or 0
•  Instead of true / false
•  Switches in parallel
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Overview of Boolean algebra
Overview of Boolean algebra
___
•  Not_ is a horizontal bar above the number
•  Example: translate (x+y+z)(xyz) to a Boolean
logic expression
–  0_ = 1
–  1 = 0
–  (x∨y∨z)∧(~x∧~y∧~z)
•  Or is a plus
•  We can define a Boolean function:
–  0+0 = 0
–  0+1 = 1
–  1+0 = 1
–  1+1 = 1
__
–  F(x,y) = (x+y)(xy) = (x∨y)∧(~x∧~y)
•  And then write a “truth table” for it:
•  And is multiplication
–  0*0 = 0
–  0*1 = 0
–  1*0 = 0
–  1*1 = 1
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x
y
F(x,y)
1
1
0
1
0
0
0
1
0
0
0
0
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Basic logic gates
Combinational circuit rules
• 
x
•  Not
x
•  And
x
y
xy
•  Or
x
y
x+y
•  Nand
x
y
xy
•  Nor
x
y
x+y
•  Xor
x
y
xÅ y
x
y
z
x
y
z
xyz
• 
x+y+z
In general, gates can be combined into
combinational circuits
Rules:
–  Never combine two input wires
–  A single input wire can be split partway and
used as input for two separate gates
–  An output wire can be used as input
–  No output of a gate can eventually feed back
into that same gate
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Converting between circuits and
Boolean expressions
Converting between circuits and
equations
•  Find the output of the following circuit
x
y
x+y
y
y
•  Find the output of the following circuit
(x+y)y
x
y
y
xy
xy
___
__
__
•  Answer: (x+y)y
•  Answer: xy
–  Or (x∨y)∧~y
–  Or ~(~x∧~y) ≡ x∨y
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Converting between circuits and
equations
Write the circuits for the following
Boolean algebraic expressions
__
a)  x+y
x
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Converting between circuits and
equations
• 
x
y
x
• 
Write the circuits for the following
Boolean
algebraic expressions
_______
b)  (x+y)x
x
y
x+y
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x+y
x+y
(x+y)x
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Writing xor using and/or/not
•  p ⊕ q ≡ (p ∨ q) ∧ ~(p ∧ q)
____
•  x ⊕ y ≡ (x + y)(xy)
x
y
x+y
xy
x
y
x⊕y
1
1
0
1
0
1
0
1
1
0
0
0
(x+y)(xy)
xy
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OR gate
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Integrated Circuits
•  Very advanced (and miniaturized)
electronic circuits
•  Least expensive way to make logic gates
in large volumes
•  Mainly consist of semiconductor devices
–  Transistors – on/off/amplify
•  Integrate a large number of transistors
onto a tiny chip (die) …
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The 7400 chip, containing four
NANDs.
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Eniac: the first computer
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Eniac’s vacuum tubes
Integrated Circuits – Chips
A back panel
of Eniac,
showing the
vacuum tubes
Upper interconnect layers on an Intel
80486 DX2 microprocessor die.
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Moore’s Law
•  The complexity of an integrated circuit,
with respect to minimum component cost,
doubles every 18 months.
•  True since the early 1970s
•  Current leading developmental constraint?
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