Ee2 chapter7 boolean_algebra

Apr 10, 2023

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IT2001PAEngineering Essentials (2/2)

Chapter 7 - Boolean Algebra

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IT2001PA Engineering Essentials (2/2)

Lesson Objectives

Upon completion of this topic, you should be able to: State the rules and functions of Boolean algebra.



Specific Objectives

Students should be able to : State the function of Boolean algebra. State the 9 equalities of Boolean algebra. State the Commutative Law. State the Associative Law. State the Distributive Law.

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Introduction

Boolean Algebra is a set of algebraic rules, named after mathematician George Boole, in which TRUE and FALSE are equated to ‘0’ and ‘1’.

Boolean algebra includes a series of operators (AND, OR, NOT, NAND (NOT AND), NOR, and XOR (exclusive OR)), which can be used to manipulate TRUE and FALSE values.

It is the basis of computer logic because the truth values can be directly associated with bits.

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Boolean Algebra Laws

Frequently, a Boolean expression is not in its simplest form.

Boolean expressions can be simplified, but we need identities or laws that apply to Boolean algebra instead of regular algebra.

These identities can be applied to single Boolean variables as well as Boolean expressions.

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A(B+C) = A*B + A*C

(A+B)*(C+D) = A*C + A*D + B*C + B*D

Commutative LawA + B = B + A

A*B = B*AAssociative Law

A+B+C = A+(B+C) = (A+B)+C = B+(A+C)

A*B*C = A*(B*C) = (A*B)*C = B*(A*C)

Distributive Law

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Commutative Law The commutative law allows the change in position

(reordering) of an ANDed or ORed variable.

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Associative Law The Associative Law allows the regrouping of

variables.

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Distributive Law

The Distributive Law shows how OR distributes over AND and vice versa.

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Boolean Algebra Theorems

Boolean theorems can help to simplify logic expression and logic circuits.

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Boolean Algebra Equalities or Identities

(j) A + AB = A + B

A + AB = A + B

The variable A may represent an expression containing more than one variable.For instance, XY(XY)

Let A = XY, then XY(XY) = A . A = 0

(a) A . 0 = 0

(b) A . 1 = A

(c) A . A = A

(d) A . A = 0

(e) A + 0 = A

(f) A + 1 = 1

(g) A + A = A

(h) A + A = 1

(i) A = A

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A*0 = 0 A*1 = AA0

A1X = A*0 = 0 X = A*1 = A

Anything ANDed with a 0 is

• equal to 0.

Anything ANDed with a 1 is

• equal to itself.

A*A = A A*A = 0AA

A1X = A*A = A X = A*A = 0

Anything ANDed with itself is


Anything ANDed with its own

• complement equals 0.

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A+0 = A A+1 = 1A0

A1X = A+0 = A X = A+1 = 1

Anything ORed with a 0 is


Anything ORed with a 1 is

• equal to 1.

A+A = A A+A = 0AA

A1X = A+A = A X = A+A = 1

Anything ORed with itself is


Anything ORed with its own

• complement equals 1.

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A = A

A X = A = A

A variable that is complemented twice will

• return to its original

A

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Elimination Law

Equivalence is demonstrated by showing the truth table derived from the expression on the left side of the equation matches that on the right side.

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