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ECE- 1551
DIGITAL LOGIC
LECTURE 6: BOOLEAN ALGEBRA
Assistant Prof. Fareena Saqib
Florida Institute of Technology
Fall 2016, 02/01/2016
Recap
 Digital logic gates
 Assignment is due before next class.
Agenda
 Chapter 2: Boolean Algebra
 Introduction


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Motivation: Understand the relationship between Boolean logic and digital computer circuits.
Review of Boolean Algebra
Transformation of Logic Gates
Learn how to design simple logic circuits.
Motivation: Combinational Logic
 Description
 Language
 Boolean algebra
 Truth table
 Schematic Diagram
 Inputs, Gates, Nets, Outputs
 Goal
 Validity: correctness, turnaround time
 Performance: power, timing, cost
 Testability: yield, diagnosis, robustness
Motivation: Combinational Logic vs. Boolean Algebra Expression
a
b
c
d
a·b
a·b + c·d
c·d
y=e·(a·b+c·d)
e
Schematic Diagram:
5 primary inputs, 1 primary output
4 components (gates)
9 signal nets
Boolean Algebra:
5 literals
4 operators
Obj: min #terms
min #literals
Schematic Diagram vs. Boolean Expression
 Boolean Expression: #literals, #operators
 Schematic Diagram: #gates, #nets, #pins
Some Definitions
 Complement: variable with a bar over it
A, B, C
 Literal: variable or its complement
A, A, B, B, C, C
 Implicant: product of literals
ABC, AC, BC
 Minterm: product that includes all input variables
ABC, ABC, ABC
 Maxterm: sum that includes all input variables
(A+B+C), (A+B+C), (A+B+C)
Digital Discipline: Binary Values
 Typically consider only two discrete values:
 1’s and 0’s
 1, TRUE, HIGH
 0, FALSE, LOW
 1 and 0 can be represented by specific voltage levels, rotating gears, fluid levels,
etc.
 Digital circuits usually depend on specific voltage levels to represent 1 and 0
 Bit: Binary digit
George Boole, 1815 - 1864
 In the latter part of the nineteenth century,
George Boole incensed philosophers and
mathematicians alike when he suggested that
logical thought could be represented through
mathematical equations.
• Taught himself mathematics and joined the
faculty of Queen’s College in Ireland.
• Wrote An Investigation of the Laws of Thought
(1854)
• Introduced binary variables
• Introduced the three fundamental logic operations:
AND, OR, and NOT.
• Computers, as we know them today, are
implementations of Boole’s Laws of Thought.
• John Atanasoff and Claude Shannon were among the
first to see this connection.
Review of Boolean Algebra
Let B be a nonempty set with two binary operations, a unary operation `, and two distinct elements 0 and
1. Then B is called a Boolean algebra if the following axioms hold.
1. Closure: A set S is closed with respect to a binary operator if, for every pair of elements of S, the
binary operator specifies a rule for obtaining a unique element of S.
 Set of natural numbers N = {1,2,3,4,….} is closed with respect to the binary operator + by the rules of
arithmetic addition.
 N is not closed with respect to binary operator -.
2. Associative law: A binary operator + on a set S is associative when
a) (a+b)+c = a+(b+c) for all a,b,c ∈ S or
b) (a * b) * c = a * (b * c) for all a,b,c ∈ S
3. Commutative laws: A binary operator on a set S is Commutative when
a) a+b=b+a or
b) a*b=b*a
Review of Boolean Algebra
1. Distributive laws: If * and . Are two binary operators on a set S,
a) + Is said to be distributive over . Whenever:
b) or . Is said to be distributive over + Whenever :
a+(b·c)=(a+b)·(a+c),
a·(b+c)=a·b+a·c
2. Identity laws: A set is said to have an identity element with respect to an operator
a) a+0=a,
b) a·1=a
3. Inverse /Complement laws:
a)
a+a’=1,
b)
a·a’=0
Switching Algebra (A subset of Boolean Algebra)
 Boolean Algebra: Each variable may have multiple values.
 Switching Algebra: Each variable can be either 1 or 0. The constraint simplifies the
derivations.
Boolean Algebra
Switching Algebra
BB
Two Level Logic
Switching Algebra
 Two Level Logic: Sum of products, or product of sums

e.g. ab + a’c + a’b’, (a’+c )(a+b’)(a+b+c’)
 Multiple Level Logic: Many layers of two level logic with some inverters,

e.g. (((a+bc)’+ab’)+b’c+c’d)’bc+c’e
Features of Digital Logic Design
 Multiple Outputs
 Don’t care sets
Handy Tools:
 Simplification Theorm
 DeMorgan’s Law: Complements
 Truth Table
 Minterm and Maxterm
 Sum of Products (SOP) and Products of Sum (POS)
 Karnaugh Map (single output, two level logic)
Review of Boolean algebra and switching functions
AND, OR, NOT
AND A B
0 0
0 1
1 0
1 1
A
1
A
0
OR
Y
0
0
0
1
AB
0 0
0 1
1 0
1 1
NOT A Y
0 1
1 0
Y
0
1
1
1
A
A
1
1
0
A
0
A
0 dominates in AND
0 blocks the output
1 passes signal A
1 dominates in OR
1 blocks the output
0 passes signal A
Review of Boolean algebra and switching functions
2. Associativity
(A+B) + C = A + (B+C)
C
A
B
A
B
C
(AB)C = A(BC)
C
A
B
A
B
C
Review of Boolean algebra and switching functions
4. Identity
A* 1 =A
A*0=0
A+ 1= 1
A+0=A
5. Complement
A + A’ = 1
A * A’ = 0
6. Distributive Law
A(B+C) = AB + AC
A+BC = (A+B)(A+C)
A
B
C
A
B
A
C
A
B
C
A
B
A
C
Basic Theorems and Properties of Boolean Algebra
Basic Theorems and Properties of Boolean Algebra
Proof of above mentioned theorems using Huntington Postulates:
1. (a) The structure is closed with respect to the operator +.
(b) The structure is closed with respect to the operator * .
2. (a) The element 0 is an identity element with respect to +; that is, x + 0 =0 + x = x.
(b) The element 1 is an identity element with respect to * ; that is, x * 1 = 1 * x = x.
3. (a) The structure is commutative with respect to +; that is, x + y = y + x.
(b) The structure is commutative with respect to * ; that is, x * y = y * x.
4. (a) The operator # is distributive over +; that is, x * (y + z) = (x * y) + (x * z).
(b) The operator + is distributive over * ; that is, x + (y * z) = (x + y) * (x + z).
5. For every element x ∈ B, there exists an element x’ ∈ B (called the complement of x)
such that (a) x + x’ = 1 and (b) x * x’ = 0.
6. There exist at least two elements x, y ∈ B such that x ≠ y.
Next Class – Reading Assignment
 Section 2.5: Boolean Functions
 De-Morgans Law
 Operator Precedence
 Boolean functions simplification.
 Section 2.6: Canonical and Standard Forms
 Discuss Minterms and Maxterms