Ee2 chapter7 boolean_algebra

IT2001PA
Engineering Essentials (2/2)

Chapter 7 - Boolean Algebra

Lecturer Name
lecturer_email@ite.edu.sg
Nov 20, 2012
Contact Number


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

IT2001PA Engineering Essentials (2/2) 2


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.

IT2001PA Engineering Essentials (2/2)


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.



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.



Commutative Law
A+B=B+A

A*B = B*A
Associative 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

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



Commutative Law
 The commutative law allows the change in position
(reordering) of an ANDed or ORed variable.



Associative Law
 The Associative Law allows the regrouping of
variables.



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



Boolean Algebra Theorems
 Boolean theorems can help to simplify logic expression
and logic circuits.



Boolean Algebra Equalities or Identities
(a) A . 0 =0 (j) A + AB = A + B
(b) A . 1 =A A + AB = A + B
(c) A . A =A
(d) A . A =0
The variable A may represent
(e) A + 0 = A an expression containing more
(f) A + 1 = 1 than one variable.
For instance, XY(XY)
(g) A + A = A
(h) A + A = 1 Let A = XY,
(i) A =A then XY(XY) = A . A = 0



A*0 = 0 A*1 = A
A A
X = A*0 = 0 X = A*1 = A
0 1
Anything ANDed with a 0 is Anything ANDed with a 1 is
• equal to 0. • equal to itself.

A*A = A A*A = 0
A A
A X = A*A = A 1 X = A*A = 0
Anything ANDed with itself is Anything ANDed with its own
• equal to itself. • complement equals 0.


A+0 = A A+1 = 1
A A
X = A+0 = A X = A+1 = 1
0 1
Anything ORed with a 0 is Anything ORed with a 1 is
• equal to itself. • equal to 1.

A+A = A A+A = 0
A A
A X = A+A = A 1 X = A+A = 1
Anything ORed with itself is Anything ORed with its own
• equal to itself. • complement equals 1.



A=A

A
A X=A=A

A variable that is complemented twice will
• return to its original



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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