EE210: Switching Systems
Lecture 5: NAND, NOR and XOR Gates,
Simplification of Algebraic Expressions
Prof. YingLi Tian
Sept. 19, 2016
Department of Electrical Engineering
The City College of New York
The City University of New York (CUNY)
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Definition of Switching Algebra
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OR (written as +)
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AND (written as · or simply two variables
catenated)
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a + b (read a OR b) is 1 if a = 1 or b = 1 or
both.
a · b = ab (read a AND b) is 1 if and only if a
= 1 and b = 1.
NOT (written´)
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a´ (read NOT a) is 1 if and only if a = 0.
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OR, AND, NOT Gates
A gate is a circuit with one output that implements one of the basic
functions such as OR and AND, but the input can be 2, 3, 4 or 8.
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OR, AND, NOT Gates
A gate is a circuit with one output that implements one of the basic
functions such as OR and AND, but the # of inputs can be 2, 3, 4 or 8.
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NAND and NOR Gates
DeMorgan:
P11a: (a + b)´ = a´ b´
P11b: (ab)´ = a´ + b´
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NAND and NOR Gates
OR
NOR
NOR AND NAND NAND
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NOT, AND, OR gates from NAND
Gates
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NAND Gate Logical Design
f = x´y + xy´ + xz
AND
OR gates
NAND
gates
Function Implementation Example
f = wx(y+z) + x´y
(Example 2.12, page 61)
1. Use AND and OR gates
2. Use NAND gates
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Function Implementation Example
f = wx(y+z) + x´y
(Example 2.12, page 61)
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Exclusive-OR and Exclusive-NOR Gates
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Ex-OR: The output is 1 if the inputs are different.
Ex-NOR: The output is 1 if the inputs are same.
EX-OR and Ex-NOR gates can only have 2
inputs.
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Truth Table of NAND, NOR, ExOR, and Ex-NOR
A
B
AND
NAND OR
NOR
Ex-OR Ex-NOR
0
1
0
1
0
0
1
1
0
0
0
1
1
1
1
0
1
0
0
0
0
1
1
0
0
1
1
1
1
0
0
1
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Simplification of Algebraic Expressions
Example in Lecture 4:
(1) x´yz´ + x´yz + xy´z´ + xy´z + xyz
(2) x´y + xy´ + xyz
(3) x´y + xy´ + xz
(4) x´y + xy´ + yz
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Simplification of Algebraic Expressions
Example in Lecture 4:
f (A, B, C) = A´B´C + A´BC´ + A´BC + AB´C´+ AB´C
= A´B´C + A´B + AB´
= A´C + A´B + AB´
= B´C + A´B + AB´
f ´(A, B, C) = A´B´C´ + ABC´ + ABC
= A´B´C´ + AB
See page56 for details.
P8a: a (b + c) = ab + ac
P9a: ab + ab´ = a
P10a: a + a´ b = a + b
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Simplification of Algebraic Expressions
Example 2.16, page 66 (3 terms, 6 literals)
f = wx + wxy + w´yz + w´y´z + w´xyz´
P8a: a (b + c) = ab + ac
P9a: ab + ab´ = a
P10a: a + a´ b = a + b
P12a: a + ab = a
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Simplification of Algebraic Expressions
Example 2.16, page 66 (3 terms, 6 literals)
f = wx + wxy + w´yz + w´y´z + w´xyz´
= (wx + wxy) + (w´yz + w´y´z ) + w´xyz´
= wx + w´z + w´xyz´
P8a: a (b + c) = ab + ac
= wx + w´(z + xyz´ ) -----P10a
P9a: ab + ab´ = a
= wx + w´(z + xy)
= wx + w´z + w´xy
P10a: a + a´ b = a + b
= w´z + x(w + w´y) -----P10a
P12a: a + ab = a
= w´z + wx + xy ---done!
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Consensus
For any two product terms where exactly one variable appears
uncomplemented in one and complemented in the other, the
consensus is defined as the product of the remaining literals.
If no such variable exists or if more than one such variable
exists, then the consensus is undefined. If we write one term
as at1 and the second as a´t2 (where t1 and t2 represent product
terms), then, the consensus is defined.
at1 ¢ a´t2 = t1t2
Examples:
ab´c ¢ a´d = b´cd
ab´c ¢ a´cd= b´cd
abc´ ¢ bcd´ = abd´
b´c´d´ ¢ b´cd ´= b´d´
abc´ ¢ bc´d= Undefined – no variable
a´ bd ¢ ab´cd= Undefined – two variables
Property -- Consensus
The consensus term is redundant and can be removed:
P13a. at1 + a´t2 + t1t2 = at1 + a´t2
P13b. (a + t1)(a´ + t2)(t1 + t2)= (a + t1)(a´ + t2)
P9a: ab + ab´ = a
P12a: a + ab = a
P13a. at1 + a´t2 = (at1 + at1t2 ) + (a´t2 + a´t1t2 )
= at1 + a´t2 + (at1t2 + a´t1t2 ) = at1 + a´t2 + t1t2
Common Mistake:
ab + b´c + ac = ab + b´c ≠ ac
Reduce functions by applying --consensus
The consensus term is redundant and can be removed:
P10a: a + a´ b = a + b
Example 2.20:
f = a´b´c´ + a´bc´ + a´bc + ab´c´ = a´c´ + a´bc + ab´c´
= a´(c´ + bc) + ab´c´ = a´(c´ + b) + ab´c´
= a´c´ + a´b + ab´c´ = a´b + c´ (a´ + ab´)
= a´b + a´c´ + b´c´ = a´b + b´c´
Consensus, removed, a´b ¢ b´c´ = a´c´
Reduce functions by applying --consensus
The consensus term is redundant and can be removed:
Example 2.22: Reduce f to 3 terms with 6 literals (use
consensus):
f = c´d´+ ac´ + ad + bd´ + ab = ?
Reduce functions by applying --consensus
The consensus term is redundant and can be removed:
Example 2.22: Reduce f to 3 terms with 6 literals (use
consensus):
f = c´d´+ ac´ + ad + bd´ + ab
= c´d´ + ad + bd´
c´d´ ¢ ad = ac´
ad ¢ bd´ = ab
Announcement:
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HW2 due 9/21.
Review Chapter 2.6-2.7
Next class (Chapter 2.8-2.9, Chapter 3.1):
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Manipulation of Algebraic Functions
Boolean Algebra
Karnaugh Maps
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