DDCO Module 1

Module-1
RBT: L1, L2
Introduction to Digital Design: Binary Logic, Basic

Theorems And Properties Of Boolean Algebra,
Boolean Functions, Digital Logic Gates, Introduction,
The Map Method, Four-Variable Map, Don’t-Care
Conditions, NAND and NOR Implementation, Other
Hardware Description Language – Verilog Model of a
simple circuit.
Binary Logic
• Binary logic deals with variables that take on
two discrete values and with operations that
assume logical meaning. The two values the
variables assume may be called by different
names (true and false, yes and no, etc.), but
for our purpose, it is convenient to think in
terms of bits and assign the values 1 and 0
Definition of Binary Logic
• Binary logic consists of binary variables and a
set of logical operations.
• The variables are designated by letters of the
alphabet, such as A, B, C, x, y, z, etc., with each
variable having two and only two distinct
possible values: 1 and 0.
• There are three basic logical operations: AND,
OR, and NOT.
Definition of Binary Logic Cont..
AND: This operation is represented by a dot or
by the absence of an operator. For example, x #
y = z or xy = z is read “x AND y is equal to z.” The
logical operation AND is interpreted to mean
that z = 1 if and only if x = 1 and y = 1; otherwise
z = 0. (Remember that x, y, and z are binary
variables and can be equal either to 1 or 0, and
nothing else.) The result of the operation x # y is
z.
OR: This operation is represented by a plus sign.
For example, x + y = z is read “x OR y is equal to
z,” meaning that z = 1 if x = 1 or if y = 1 or if both
x = 1 and y = 1. If both x = 0 and y = 0, then z = 0.
NOT: This operation is represented by a prime
(sometimes by an over bar). For example, x = z
(or x = z ) is read “not x is equal to z,” meaning
that z is what x is not. In other words, if x = 1,
then z = 0, but if x = 0, then z = 1. The NOT
operation is also referred to as the complement
operation, since it changes a 1 to 0 and a 0 to 1,
i.e., the result of complementing 1 is 0, and vice
versa.
Boolean Algebra
Boolean algebra:
• Like any other deductive mathematical system, may be defined with a set of
elements, a set of operators, and a number of unproved axioms or postulates.
• A set of elements is any collection of objects, usually having a common
property. If S is a set, and x and y are certain objects, then the notation x € S
means that x is a member of the set S and y S means that y is not an element
of S.
• A set with a denumerable number of elements is specified by braces: A = {1,
2, 3, 4} indicates that the elements of set A are the numbers 1, 2, 3, and 4. A
binary operator defined on a set S of elements is a rule that assigns, to each
pair of elements from S, a unique element from S.
• As an example, consider the relation a *b = c. We say that * is a binary
operator if it specifies a rule for finding c from the pair (a, b) and also if a, b, c
H S. However, * is not a binary operator if a, b H S, and if c x S.
• The postulates of a mathematical system form
the basic assumptions from which it is
possible to deduce the rules, theorems, and
properties of the system.
• The most common postulates used to
formulate various algebraic structures are as
follows:
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.
• For example, the set of natural numbers N = {1, 2, 3, 4,
c} is closed with respect to the binary operator + by the
rules of arithmetic addition, since, for any a, b H N,
there is a unique c H N such that a + b = c.
• The set of natural numbers is not closed with respect to
the binary operator - by the rules of arithmetic
subtraction, because 2 - 3 = -1 and 2, 3 H N, but (-1) x N.
2. Associative law:
• A binary operator * on a set S is said to
be associative whenever
(x * y) * z = x * (y * z) for all x, y, z, H S
• Commutative law:
A binary operator * on a set S is said to be
commutative whenever
x * y = y * x for all x, y H S
• Identity element:
A set S is said to have an identity element with respect to a binary
operation * on S if there exists an element e belongs to S with the property that
e * x = x * e = x for every x € S
Example: The element 0 is an identity element with respect to the binary

operator + on the set of integers I = { c, -3, -2, -1, 0, 1, 2, 3, c}, since
x + 0 = 0 + x = x for any x € I
The set of natural numbers, N, has no identity element, since 0 is excluded from
the set.
• Inverse. A set S having the identity element e
with respect to a binary operator * is said to
have an inverse whenever, for every x € S,
there exists an element y € S such that
x*y=e
Example: In the set of integers, I, and the
operator +, with e = 0, the inverse of an element
a is (-a), since a + (-a) = 0.
Distributive law. If * and # are two binary operators
on a set S, * is said to be distributive over whenever
x * (y . z) = (x * y) .(x * z)
A field is an example of an algebraic structure. A
field is a set of elements, together with two binary
operators, each having properties 1 through 5 and
both operators combining to give property 6. The
set of real numbers, together with the binary
operators + and . , forms the field of real numbers.
The field of real numbers is the basis for arithmetic and ordinary
algebra. The operators and postulates have the following meanings:
• The binary operator + defines addition.
• The additive identity is 0.
• The additive inverse defines subtraction.
• The binary operator . defines multiplication.
• The multiplicative identity is 1.
• For a ≠ 0, the multiplicative inverse of a = 1/a defines division
(i.e., a . 1/a = 1 ).
• The only distributive law applicable is that of # over +:
a . (b + c) = (a .b) + (a . c)
Simplify:
1. x(x + y) = xx + xy = 0 + xy = xy.
2. x + xy = (x + x)(x + y) = 1(x + y) = x + y.
3. (x + y)(x + y) = x + xy + xy + yy = x(1 + y + y) = x.
4. xy + xz + yz = xy + xz + yz(x + x) = xy + xz + xyz + xyz = xy(1 + z) + xz(1 + y) = xy + xz.
5. (x + y)(x + z)(y + z) = (x + y)(x + z), by duality from function 4
Complement of a Function:
The complement of a function F is F and is obtained from an
interchange of 0’s for 1’s and 1’s for 0’s in the value of F. The complement of a
function may be derived algebraically through De Morgan’s theorems
Theorem 5, De Morgan
(a) (x + y) = x y
(b) (b) (xy) = x + y
DeMorgan’s theorems can be extended to three or more variables. The three‐

variable form of the first DeMorgan’s theorem is derived as follows, from
postulates and theorems
(A + B + C)’ = (A + x)’ let B + C = x
= A’x’ by theorem 5(a) (DeMorgan)
= A’(B + C)’ substitute B + C = x
= A’(B’C’) by theorem 5(a) (DeMorgan)
= A’B’C’ by theorem 4(b) (associative)
Find the complement of the functions
F1 = x’yz’ + x’y’z and F2 = x(y’z’+ yz).
By applying De Morgan’s theorems as many times as necessary, the
complements are obtained as follows:
F ‘1 = (x’yz’ + x’y’z) ‘
= (x’yz’)’(x’y’z)’
= (x + y’+ z)(x + y + z’)
F’2 = [x(y’z’ + yz)]’
= x’ + (y’z’ + yz)’
= x’ + (y’z’)’(yz)’
= x’ + (y + z)(y’+ z’)
= x’ + yz’+ y’z

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Module-1

Introduction to Digital Design: Binary Logic, Basic

Example: The element 0 is an identity element with respect to the binary

DeMorgan’s theorems can be extended to three or more variables. The three‐

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