Philadelphia University

Faculty of Information Technology

Department of Computer Science

Computer Logic Design

By

Dareen Hamoudeh

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Simplification Using Map Method

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Why map method?

• Complex algebraic expression Complex Logic gates.

• Several algebraic expressions for same function.

• Function minimization using algebraic expression is awkward no specific rules to predict each step in the manipulative process.

• Map Method:

– Provides simple, straightforward procedure in minimizing functions.

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Map method (K-map)

• Also known as: – Veitch diagram.

– Karnaugh map.

• The Diagram made up of squares , each square represents one minterm.

• Represents visual diagram of all possible ways a function may expressed in standard form.

• We will assume: the simplest algebraic expression is any one in(SOP) or (POS) that has minimum numbers of literals.

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Two Variables Map

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Two Variables Map

• There are 4 minterms for two variables, so the map consists of 4 squares one for each minterm.

• We mark 0 and 1 for each row & column designate x and y:

X: primed in row 0.

Unprimed in row 1.

y: primed in col. 0.

Unprimed in col. 1.

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Two Variables Map

• We only mark the squares whose minterm belong to the given function.

• If we have F=x.y, it is equal to m3 ,because it is = 1 when x=1 and y=1. so, we place 1 inside the square that belong to m3:

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Two Variables Map

• If we have F=x+y, then its minterms are:

X+y=X’.y+x.y’+x.y=m1+m2+m3

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Three Variables Map

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Three Variables Map • There are 8 minterms.

• Map consists of 8 squares.

• Minterms are arranged in a sequence similar to reflected code.

• Only one bit changes from 1 to 0 or from 0 to 1 in the sequence.

• There are 4 squares where each variable =1, and 4 squares where each variable =0.

• We write the variable with its letter symbol under the four squares where it is unprimed.

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Map in simplification

• Basic property for adjacent squares in the map:

– Any two adjacent squares differ by only one variable: primed in square & unprimed in the other.

– EX:

In m5 & m7 : y is primed in m5 and unprimed in m7, from postulates m5+m7= xy’z+ xyz = xy(y’+y) = xy.

Sum of minterms in adjacent squares can simplified to a single AND term with 2 literals.

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Example 1

• Simplify the Boolean function using K-map

F=x’yz + x’yz’+ xy’z’+ xy’z Solution:

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Represents x’.y

Represents x.y’

(m0 + m2) and (m4 + m6)

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• Solution:

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Four adjacent squares

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Example 2

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• Solution:

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Example 3

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• Solution:

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Most minimization example

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F= Z’

Most minimization example

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F= x’+y

Four Variables Map

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Four Variables Map

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Four Variables Map

• Like three-variable map: we minimize function using Adjacent squares property.

• In addition the map is considered to lie on surface with the top and bottom edges as well as the right and the left, for Example:

– m0 and m2 form adjacent squares.

– m3 and m11 form adjacent squares.

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Four Variables Map

• Combination of adjacent squares is easily determined:

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Example 1

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Solution:

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Example 2

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• Solution:

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Self Study & Practice

Five Variables Map

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Five Variables Map

• Number of squares = number of minterms: 25 =32 • Rows & columns are numbered in reflected code

sequence. • There are 16 squares where each variable =1, and 16

squares where each variable =0. • As it consists of 2 four-variable maps. • Each four-variable maps is recognized from the double

line in the center: – Each retains the previously defined adjacency, individually. – In addition, the center lines considered as the center of a

book, with each half of the map being a page

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Represented as 2 four-variable map

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Five Variables Map

• When the book is closed, two adjacent squares will fall one in each other, beside its four neighboring squares.

• Example: m31 is adjacent to m30,m15,m29,m23 and m27

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Five Variables Map

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Solution

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NAND and NOR Implementation

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NAND and NOR Implementation

• Digital circuits are frequently constructed with only NAND or NOR gates.

– because these gates are easier to fabricate with electronic components.

• Because of the importance of NAND and NOR in the design of digital circuits.

– rules and procedures have been developed for the conversion from Boolean functions in terms of AND, OR and NOT into equivalent NAND or NOR logic diagrams.

• NAND and NOR are called universal gates.

– because any digital system or Boolean function can be implemented with only these gates.

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NAND and NOR Implementation

• Two-level implementation is presented here.

• There are two other graphic symbols for these gates, to facilitate conversions.

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NAND and NOR Implementation

• NAND equivalent symbols:

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NAND equivalent symbols

• Consists of an AND symbol followed by small circle.

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NAND equivalent symbols:

• OR symbol preceded by small circles in all the inputs.

• It follows DeMorgan’s theorem where small circles denote complementation.

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NAND and NOR Implementation

• NOR equivalent symbols:

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NOR equivalent symbols

• Consists of an OR symbol followed by small circle.

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NOR equivalent symbols:

• AND symbol preceded by small circles in all the inputs.

• It follows DeMorgan’s theorem where small circles denote complementation.

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NAND and NOR Implementation

• One-input NAND or NOR gate:

– Inverter.

• Three different graphic symbols for inverter:

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NAND and NOR Implementation

• NAND Simple Examples:

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NAND and NOR Implementation

• NOR Simple Examples:

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