Logic gates form the basic building blocks of digital circuits and logic. The three fundamental logic gates are AND, OR, and NOT. NAND and NOR gates are also commonly used as they are universal gates that can be combined to perform all possible logic functions. Techniques for minimizing logic expressions include Karnaugh maps, which allow visualization of minterms, and the Quine-McCluskey method, which systematically finds prime implicants. Implementation of logic functions typically uses NAND or NOR gates due to their simplicity and universality.

1. Boolean Algebra

2. Logic Gates
Features of logic gates are as follows:
• The flow of digital signals is controlled by transistors in various configurations depending on
the logic family. For most purposes we can imagine that the logic gates are composed of
ideal switches with just two states: OPEN and CLOSED. The state of a switch is controlled by
a digital signal. The switch remains closed so long as a logical (1) signal is applied. A logical
(0) control signal keeps it open.
• Logic signals interact by means of gates. The three fundamental gates AND, OR, and NOT, are
named after the three fundamental operations of logic that they carry out. The AND and OR
gates each have two inputs and one output. The output state is determined by the states of the
two inputs.

3. NOT Gate
The NOT gate has a single input and a single output and its symbol
and truth table (truth table contains a table of all possible input
values and their corresponding outputs values). It performs the
operation of inversion, i.e., the output of a NOT gate takes a 1
(high) state if the input takes the 0 (low) state and vice-versa. A
circuit, which performs a logic negation, is called a NOT circuit or
inverter since it inverts the output with respect to the input.

4. NAND Gate
• The Negated AND, NOT AND or NAND gate is the opposite of the digital
AND gate, and behaves in a manner that corresponds to the opposite of
AND gate, as shown in the truth table. A LOW output results only if both
the inputs to the gate are HIGH. If one or both inputs are LOW, a HIGH
output results. The NAND gate is a universal gate in the sense that any
Boolean function can be implemented by NAND gates.
• Digital systems employing certain logic circuits take advantage of
NAND's functional completeness. In complicated logical expressions,
normally written in terms of other logic functions such as AND, OR, and
NOT, writing these in terms of NAND saves on cost, because
implementing such circuits using NAND gate yields a more compact result
than the alternatives.

5. Karnaugh Maps (K-Maps)
The Karnaugh map uses a rectangle divided into rows and
columns in such a way that any product term in the expression to be
simplified can be represented as the intersection of arow and
a column.
The rows and columns are labelled with each term in the
expression and its complement. The labels must be arranged so that
each horizontal or vertical move changes the state of one and only
one variable. A Karnaugh map is visual display of minterms
required for a sum-of-product solution.

6. Sum-of-Products Equations and Logic Circuits (SOP)
There are four possible ways to AND two input signals that are
in the complemented and un-complemented form (B, A, AB) also
called as fundamental products. Table 2.1 lists each of the
fundamental product producing high outputs.
A B Fundamental
Product
Minterms
0 1 B m1
1 1 AB m3

7. Product of Sums (POS)
The Product of Sums form represents an expression as a
product of maxterms. Each individual term in standard POS
form is called as maxterm.
F(X, Y ...) = Product (bk + Mk), where bk is 0 or 1 and Mk
is a maxterm.

8. NAND AND NOR IMPLEMENTATION
Digital circuit are more frequently constructed with
only NAND or NOR gates, rather then with AND and OR gates.
NAND ana NOR gates are easier to fabricate with electronic
components and are the universal gates used in all IC digital
logic families. NAND and NO gates are used in the design of
digital circuit, rules and procedures have been developed for
the conversions from Boolean functions given in terms of AND,
OR and NOT into equivalent NAND and NOR logic diagrams.

9. NAND IMPLEMENTATION
1. Simplify the function and express it in the sum of
product form.
2. Draw a NAND gate for each product term of the
function, that has at least two literals. The inputs to
each NAND gate are the literals of the term. The
constitutes a group of first level gats.
3. Draw single NAND gar in the second level, with inputs
coming from the outputs f first level gates

10. NOR IMPLEMENTATION
The NOR functions is the dual of the NAND function.
Because of this reasons, all procedures and rules for NOR
logic are the duals if the corresponding procedures and
rules developed for NAND logic.
The implementation of a Boolean function with nor
gates requires that the function simplified in the product of
sus form. A product of sums expression specifies group OR
gates for the s terms followed by an AND gate produce the
product. Consider the product of sum expression.
F= (A+B) (C+D)E

11. OR AND INVERT IMPLEMENTATION
The OR –NAND and NOR-OR forms perform the OE-AND-
INVERT functions. The OR-NAND form resembles the OR-
AND form, expect for the inversions done by the circle in
the NAND gate.

12. Don’t Care Conditions
In some digital systems, some inputs are never supplied,
thereby no output is visible. We call such cases "don’t care
conditions" and they are represented by X in the truth table as
depicted in Fig. 2.4(c). Whenever we see an X in the truth
table while encircling the 1s in the K-map to form the largest
group, we can assume Xs as 1s. After it has been included in
all the groups, disregard the Xs in the truth tables by assuming
them as 0s. As for Fig. 2.4(c), the SOP equation after
Karnaugh simplificationsare Y = B.

13. Quine - McCluskey Method
The Quine - McCluskey algorithm (or the method of prime
implicants) is a method used for minimization of Boolean
functions which was developed by W.V. Quine and Edward J.
McCluskey. It is functionally identical to Karnaugh mapping,
But thetabular for makes it more efficient for use incomputer
algorithms, and it alsogives a deterministic way to
check that the minimal form of a Boolean function has been
reached. It is sometimes referred to as the tabulation method.

14. The method involves two steps:
• Finding all prime implicants of the function.
• Use those prime implicants in a prime implicant chart to
find the essential prime implicantsof the function, as well as
other prime implicants that are necessary to cover the
function.
Thus, logic expressions can be represented in one of the
standard forms: SOP or POSand then simplified using K-
maps or Quine- McCluskey method.

15. Logic Minimisation
Given a truth table, it is always possible to write down a correct logic
expression simply by forming an OR of the ANDs of all input variables for which the
output is true. However, for an arbitrary truth table such a procedure could produce a
very lengthy and cumbersome expression which might be needlessly inefficient
to implement with gates.
There are several methods for simplification of Boolean logic expressions. The
process is usually called "logic minimization“ and the goal is to form a result which
is efficient. Two methods we will discussare algebraic minimization and
Karnaugh maps. For more complex expressions the Quine-McKluskey method may
be appropriate. Karnaugh maps are also limited to problems with up to 4 binary
inputs.

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