Unformatted text preview: Department of Computer and Information Science,
School of Science, IUPUI CSCI 240 Boolean Algebra
Introduction
Dale Roberts, Lecturer
Computer Science, IUPUI
Email: [email protected] Dale Roberts Boolean Algebra
We observed in our introduction that early in the development
We
of computer hardware, a decision was made to use binary
circuits because it greatly simplified the electronic circuit
design.
design.
In order to work with binary circuits, it is helpful to have a
In
conceptual framework to manipulate the circuits algebraically,
building only the final “most simple” result.
building
George Boole (18131864) developed a mathematical structure
George
to deal with binary operations with just two values. Today, we
call these structures Boolean Algebras.
Boolean
Dale Roberts Boolean Algebra Defined
A Boolean Algebra B is defined as a 5tuple {B, +, *, ’, 0, 1}
Boolean
’,
+ and * are binary operators,’ is a unary operator.
and
binary
unary
The following axioms must hold for any elements a, b, c ∈ {0,1}
The Axiom #1: Closure If a and b are elements of B, (a + b) and (a * b) are in B.
If Axiom #2: Cardinality
There are at least two elements a and b in B such that a != b.
There Axiom #3: Commutative
If a and b are elements of B
If
(a + b) = (b + a), and (a * b) = (b * a) Dale Roberts Axioms
Axiom #4: Associative If a and b are elements of B
(a + b) + c = a + (b + c), and (a * b) * c = a * (b * c) Axiom #5: Identity Element B has identity elements with respect to + and *
0 is the identity element for +, and 1 is the identity element for *
a + 0 = a and a * 1 = a Axiom #6: Distributive
* is distributive over + and + is distributive over *
is
a * (b + c) = (a * b) + (a * c), and a + (b * c) = (a + b) * (a + c) Axiom #7: Complement Element
For every a in B there is an element a' in B such that
For
a + a' = 1, and a * a' = 0 Dale Roberts Terminology
Element 0 is called “FALSE”.
Element 1 is called “TRUE”.
‘+’ operation “OR”,‘*’ operation “AND” and ’ operation
+’
“NOT”.
Juxtaposition implies * operation: ab = a * b
ab
Operator order of precedence is: (), ’, *, +.
Operator
’,
a+bc = a+(b*c) ≠ (a+b)*c
a+bc
(a+b)*c
ab’ = a(b’) ≠ (a*b)’ Single Bit Boolean Algebra(1’ = 0 and 0’ = 1)
+
0 0
0 1
1 *
0 0
0 1
0 1 1 1 1 0 1
Dale Roberts Proof by Truth Table
Consider the distributive theorem: a + (b * c) = (a + b)*(a + c).
Consider
(b
Is it true for a two bit Boolean Algebra?
Is
Can prove using a truth table. How many possible
Can
combinations of a, b, and c are there?
and
Three variables, each with two values: 2*2*2 = 23 = 8
Three
a b c b*c a+(b*c) a+b a+c (a+b)*(a+c
) 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 Dale Roberts nbit Boolean Algebra
Single bit Boolean Algebra can be extended to nbit Boolean
Single
nbit
Algebra by define sum(+), product(*) and complement(‘) as bitand
wise operations
Let a = 1101010, b = 1011011
a + b = 1101010 + 1011011 = 1111011
1101010
a * b = 1101010 * 1011011 = 1001010
1101010
a’ = 1101010’ = 0010101 Principle of Duality The dual of a statement S is obtained by interchanging * and +; 0
and 1.
and
Dual of (a*1)*(0+a’) = 0 is (a+0)+(1*a’) = 1
Dual
Dual of any theorem in a Boolean Algebra is also a theorem.
Dual
This is called the Principle of Duality.
Principle
Dale Roberts Named Theorems
All of the following theorems can be proven based on the axioms.
All
They are used so often that they have names.
They
Idempotent a+a=a a*a=a Boundedness a+1=1 a*0=0 Absorption a + (a*b) = a a*(a+b) = a Associative (a+b)+c=a+(b+c) (a*b)*c=a*(b*c) The theorems can be proven for a twobit Boolean Algebra using
The
a truth table, but you must use the axioms to prove it in
general for all Boolean Algebras.
general
Dale Roberts More Named Theorems
Involution (a’)’ = a DeMorgan’s (a+b)’ = a’ * b’ (a*b)’=a’ + b’
DeMorgan’s Laws are particularly important in circuit design. It
DeMorgan’s
says that you can get rid of a complemented output by
complementing all the inputs and changing ANDs to ORs. (More
about circuits coming up…)
about Dale Roberts Proof using Theorems
Use the properties of Boolean Algebra to reduce (x +
Use
y)(x + x) to x. Warning, make sure you use the laws
y)(x
precisely.
precisely.
(x + y)(x + x)
(x Given (x + y)x
(x Idempotent x(x + y)
x(x Commutative x Absorption Unlike truth tables, proofs using Theorems are valid for any
boolean algebra, but just bits.
Dale Roberts Sources
Lipschutz, Discrete Mathematics
Mowle, A Systematic Approach to Digital Logic
Design Dale Roberts ...
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Full Document
 Spring '10
 Won
 Computer Science, Logic, Boolean Algebra, Propositional calculus, Binary operation, Dale Roberts

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