Complement of a Function

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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
Reading Assignment Chapter 2 in Logic and
Computer Design Fundamentals, 4th Edition by Mano

• Complement of a Function
• A function F has the exact opposite truth table
  as function F.

Example Find truth table for F and for F for
the function below.

• Canonical and Standard forms
• Boolean functions are commonly expressed using
  the following forms
• Canonical forms
• Sum of minterms
• Product of maxterms
• Standard forms
• Sum of products
• Product of sums

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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
Minterm (also called a standard product) A
minterm is a term containing all n variables
(complemented or uncomplemented) ANDed together.
Example f(A,B) has 4 possible minterms. List
them.
Each minterm represents one n-bit word
where Primed variable ? 0 Unprimed variable ?
1 Minterm designation for function f(x,y,z)
the input combination 000 represents minterm
xyz and is designated m0.
Example Show all 8 possible minterms and the
shorthand designations for f(x, y, z).
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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
Key Point A Boolean function F may be
represented by a sum (ORed together) of its
minterms. They represent the input combinations
needed to yield F 1. So minterms represent the
1s in the truth table for F.
Example Pick a truth table for some function
f(x,y,z) and represent f as a sum of minterms.
Maxterm (also called a standard sum) A maxterm
is a term containing all n variables
(complemented or uncomplemented) ORed together.
Example f(A,B) has 4 possible maxterms. List
them.
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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
Each maxterm represents one n-bit word
where Primed variable ? 1 (note that
this is opposite of the notation used for
minterms) Unprimed variable ? 0 Maxterm
designation for function f(x,y,z) the input
combination 000 represents maxterm (x y z)
and is designated M0.
Example Show all 8 possible maxterms and the
shorthand designations for f(x, y, z).
Key Point A Boolean function F may be
represented by a product (ANDed together) of its
maxterms. They represent the input combinations
needed to yield F 0. So maxterms represent the
0s in the truth table for F.
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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
Example Pick a truth table for some function
f(x,y,z) and represent f as a product of maxterms.
Relationship between minterms and maxterms Show
that and
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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
Conversion between forms Since minterms represent
where F 1 and maxterms represent where F 0,
all terms are either minterms or maxterms. So if
F is expressed as a sum of minterms, then F is a
product of the maxterms (the terms that were not
minterms). So it is simple to convert between
forms.
Example Convert to the other canonical form
1. F(A, B) ?(0, 1) 2. F(x, y, z) ?(0, 1)
3. F(x, y, z) ?(4, 5, 6) 4. F(a, b, c, d,
e) ?(0-4, 8, 13-18)
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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
Conversion to sum of minterms or product of
maxterms forms from other forms Possible
approaches include Boolean algebra and truth
tables.
Example Represent each function below as a sum
of minterms 1. F(A, B) A 2. F(x, y, z)
xy z Examples Represent each function
below as a product of maxterms 1. F(A, B) AB
AB 2. F(x, y, z) x y
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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
Standard Forms Canonical forms are not minimized
and are not useful for many circuit
implementations. Standard forms are more useful.
Functions are typically minimized into one of the
two standard forms 1. Sum of Products (SOP)
F sum of ANDed terms (but not necessarily
minterms) 2. Product of Sums (POS) F
product of ORed terms (but not necessarily
maxterms)
Example List several examples of SOP
expressions.
Example List several examples of POS
expressions.
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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
Example Function F(A,B,C) has the following
truth table. Express F in each of the following
forms 1. Sum of minterms 2. Product of
maxterms 3. Minimal SOP 4. Minimal POS
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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
Standard Forms 2-Level Implementations Standard
forms are referred to as 2-level
implementations because they can be implemented
with two levels gates (and thus only two gate
delays). Note that this does not include initial
inverters.
Example Implement a SOP expression using logic
gates to illustrate that it is a 2-level
implementation.
Example Implement a POS expression using logic
gates to illustrate that it is a 2-level
implementation.
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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
Non-standard forms 4 commonly used forms have
been covered (sum of minterms, product of
maxterms, SOP, and POS). These forms will be
used commonly throughout the course. There are,
however, other forms.
Example List examples of non-standard
expressions and implement at least one of them
using logic gates.
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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
Basic functions/gates and their truth tables We
have previously defined two functions with two or
more inputs AND and OR. How many possible
2-input logic functions could be defined
(consider the diagram shown below)? How many
correspond to actual gates (commercially
available)? List possible truth tables
6 commonly defined 2-input logic functions/gates
1. AND 4. NOR 2. OR 5. XOR
(Exclusive-OR) 3. NAND 6. XNOR (Exclusive-NOR
or Equivalence)
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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
NAND Show logic symbol, truth table, and logic
expressions
NOR Show logic symbol, truth table, and logic
expressions
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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
XOR Show logic symbol, truth table, and logic
expressions
XNOR Show logic symbol, truth table, and logic
expressions
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Lecture 3 EGR 270 Fundamentals of
Computer Engineering
Other Logic Symbols