EGR 277 Digital Logic

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Lecture 2 EGR 270 Fundamentals of
Computer Engineering
Reading Assignment Chapter 2 in Logic and
Computer Design Fundamentals, 4th Edition by Mano
Chapter 2 - Boolean Algebra - comparison to
regular algebra   Any algebra is built upon 1)
A set of elements 2) A set of operators 3) A
set of postulates Boolean Algebra is built
upon 1) A set of elements 0, 1 2) A set of
operators , Define these in class 3) A
set of postulates the Huntington Postulates are
the most common
Huntington Postulates The following 6
postulates, along with the set of elements and
set of operators shown above, uniquely and
completely define Boolean algebra. 1) Closure
for the operations , - Discuss
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Lecture 2 EGR 270 Fundamentals of
Computer Engineering
2) Two identity elements - Illustrate by
considering all possible values for x A)
0 0 x x 0 x B) 1 1 x x
1 x 3) Commutative Laws - Illustrate by
considering all possible values for x and y
A) x y y x B) xy yx 4)
Distributive Laws - Prove by truth table
A) x (y z) xy xz B) x yz (x
y) (x z)
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Lecture 2 EGR 270 Fundamentals of
Computer Engineering
5) Existence of a Complement - Illustrate by
considering all possible values for x
Define by the following truth table
A) x x 1 B) x x 0 6)
At least two non-equal elements 0, 1 - Discuss
Common Theorems Boolean algebra has already been
completely defined. Additional theorems are also
often used, not because they are required, but
because they are useful. Some of the most common
theorems are shown below. Note that each theorem
could be formally proven using the postulates. 1)
Idempotency (same power) A) x x x
Prove this using the postulates B) x x
x
Example Show related examples using this
theorem.
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Lecture 2 EGR 270 Fundamentals of
Computer Engineering
2) (no name) Discuss A) x 1 1
B) x 0 0 3) Involution Discuss x
x 4) Associative Laws Discuss (show
logic gate application) A) x (y z)
(x y) z B) x(yz) (xy)z 5)
DeMorgans Theorems - Prove 5A by truth table
A) B)
Example Show related examples using DeMorgans
theorem.
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Lecture 2 EGR 270 Fundamentals of
Computer Engineering
6) Absorption A) x xy x B) x
(xy) x
Example Show related examples using this
theorem.
7) (no name) A) x xy x y
B) x (x y) xy
Example Show related examples using this
theorem.
8) Concensus A) xy xz yz xy
xz B) (x y)(x z)( y z) (x
y)(x z)
Example Show related examples using this
theorem.
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Lecture 2 EGR 270 Fundamentals of
Computer Engineering
Order of operations
Example f a?bc?d Note spacing is often
used to make it clearer f ab cd
Boolean Functions Simplifying Boolean functions
corresponds to minimizing the amount of circuitry
(logic gates) to be used. Truth table ? Boolean
function ? minimized with Boolean algebra
? implement with logic
circuits Minimizing Boolean functions No specific
rules. In general we use Boolean algebra
(postulates and theorems) to reduce the number of
terms, literals, logic gates, or ICs. Literal
a primed (complemented) or unprimed variable. In
counting literals, we count all occurrences of
each literal.
Example How many literals are in the expression
f ab ac bcd ? (Answer 7)
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Lecture 2 EGR 270 Fundamentals of
Computer Engineering
Examples Minimize the following Boolean
functions  1) F AB A(B C) B(B
C)   2) F AB(C BD) AB 3)
F(A,B,C,D) A ABC C
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Lecture 2 EGR 270 Fundamentals of
Computer Engineering
Examples Minimize the following Boolean
functions (continued)  4) F (xy) z
  5)
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Lecture 2 EGR 270 Fundamentals of
Computer Engineering
Examples Minimize the following Boolean
functions (continued)  6) f(x,y,z) xy(z
yx) yz   7)