1. ©2004 Brooks/Cole
FIGURES FOR
CHAPTER 2
Boolean Algebra
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This chapter in the book includes:
Objectives
Study Guide
2.1 Introduction
2.2 Basic Operations
2.3 Boolean Expressions and Truth Tables
2.4 Basic Theorems
2.5 Commutative, Associative, and Distributive Laws
2.6 Simplification Theorems
2.7 Multiplying Out and Factoring
2.8 DeMorgan’s Laws
Problems
Laws and Theorems of Boolean Algebra

2. ©2004 Brooks/Cole
The basic mathematics needed for the
study of the logic design of digital systems is
Boolean algebra. Boolean algebra has many
other applications including set theory and
mathematical logic, but we will restrict ourselves
to its application to switching circuits in this
text. Because all of the switching devices which
we will use are essentially two-state devices
(such as a transistor with high or low output
voltage), we will study the special case of
Boolean algebra in which all of the variables
assume only one of two values. This two-valued
Boolean algebra is often referred to as
2.1 Introduction
Boolean Algebra

3. ©2004 Brooks/Cole
We will use a Boolean variable, such as X
or Y. to represent the input or output of a
switching circuit. We will assume that each of
these variables can take on only two different
values. The symbols "o" and "1" are used to
represent these two different values. Thus, if Xis
a Boolean (switching) variable, then either X= o
or X = 1.
The symbols "o" and "1" used in Boolean
algebra do not have a numeric value; instead
they represent two different states in a logic
circuit and are the two values of a switching
variable. In a logic gate circuit, o (usually)
represents a range of low voltages, and 1
represents a range of high voltages. In a switch

4. ©2004 Brooks/Cole
The basic operations of Boolean algebra
are AND, OR, and complement (or inverse). The
complement of 0 is 1, and the complement of 1
is 0. Symbolically, we write
2.2 Basic Operations
0' = 1 and 1'
= 0
X' = 1 if X = 0 and X' =
0 if X = 1
where the prime (') denotes
complementation. If X is a switching variable,

5. ©2004 Brooks/Cole
An alternate name for complementation is
inversion, and the electronic circuit which forms
the inverse of X is referred to as an inverter.
Symbolically, we represent an inverter by
Section 2.2, p. 34

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where the circle at the output indicates
inversion. If a logic 0 corresponds to a low
voltage and a logic 1 corresponds to a high
voltage, a low voltage at the inverter input
produces a high voltage at the output and vice
versa. Complementation is sometimes referred
to as the NOT operation because X = 1 if X is
not equal to 0.
The AND operation can be defined as
follows:

7. ©2004 Brooks/Cole
Section 2.2, p. 34

8. ©2004 Brooks/Cole
Section 2.2, p. 34
Note that C = 1 if (if and only if) A and B
are both 1, hence, the name AND operation. A
logic gate which performs the AND operation is
represented by

9. ©2004 Brooks/Cole
Section 2.2, p. 35
The dot symbol (.) is frequently omitted in
a Boolean expression, and we will usually write
AB instead of A ~ B. The AND operation is also
referred to as logical (or Boolean) multiplication.
The OR operation can be defined as
follows:

10. ©2004 Brooks/Cole
Section 2.2, p. 35
where" + " denotes OR. If we write C = A
+ B, then given the values of A and B, we can
determine C from the following table:

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Section 2.2, p. 35
Note that C = 1 if A or B (or both) is 1,
hence, the name OR operation. This type of OR
operation is sometimes referred to as inclusive-
OR. A logic gate which performs the OR
operation is represented by
The OR operation is also referred to as
logical (or Boolean) addition. Electronic circuits
which realize inverters and AND and OR gates
are described in Appendix A.

12. ©2004 Brooks/Cole
Section 2.2, p. 35
Next, we will apply switching algebra to
describe circuits containing switches. We will
label each switch with a variable. If switch X is
open, then we will define the value of X to be 0;
if switch X is closed, then we will define the
value of X to be 1.

13. ©2004 Brooks/Cole
Section 2.2, p. 35
Now consider a circuit composed of two
switches in a series. We will define the
transmission between the terminals as T = 0 if
there is an open circuit between the terminals
and T = 1 if there is a closed circuit between the
terminals.

14. ©2004 Brooks/Cole
Section 2.2, p. 35
Now we have a closed circuit between
terminals 1 and 2 (T = 1) iff (if and only if)
switch A is closed and switch B is close.. Stating
this algebraically,
Next consider a circuit composed of two
switches in parallel.

15. ©2004 Brooks/Cole
In this case, we have a closed circuit
between terminals 1 and 2 if switch A is closed
or switch B is closed. Using the same
convention for defining variables as above, an
equation which describes the behavior of this
circuit is
Thus, switches in a series perform the
AND operation and switches in parallel perform
the OR operation.

16. ©2004 Brooks/Cole
Boolean expressions are formed by
application of the basic operations to one or
more variables or constants. The simplest
expressions consist of a single constant or
variable, such as o, X. or Y. More complicated
expressions are formed by combining two or
more other expressions using AND or OR, or by
complementing another expression. Examples
of expressions are
2.3 Boolean Expressions and Truth
Tables
AB’+C
(2-1)
[AC+D]’+BE
(2-2)

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Parentheses are added as needed to
specify the order in which the operations are
performed. When parentheses are omitted,
complementation is performed first followed by
AND and then OR. Thus in Expression (2-1), B' is
formed first, then AG', and finally AB' + C.
Each expression corresponds directly to a
circuit of logic gates. Figure 2-1 gives the
circuits for Expressions (2-1) and (2-2).

18. ©2004 Brooks/Cole
Figure 2-1: Circuits for Expressions (2-1)
and (2-2)

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An expression is evaluated by substituting
a value of 0 or 1 for each variable. If A = B = C =
1 and D = E = 0, the value of Expression (2-2) is
Each appearance of a variable or its
complement in an expression will be referred to
as a literal. Thus, the following expression,
which has three variables, has 10 literals:
ab'c + a'b + a'bc' + b'c'

20. ©2004 Brooks/Cole
When an expression is realized using logic
gates, each literal in the expression corresponds
to a gate input.
A truth table (also called a table of combinations)
specifies the values of a Boolean expression for every
possible combination of values of the variables in the
expression. The name truth table conies from a similar
table which is used in symbolic logic to list the truth or
falsity of a statement under all possible conditions. We
can use a truth table to specify the output values for a
circuit of logic gates in terms of the values of the input
variables. The output of the circuit in Figure 2-2(a) is F =
A' ± B. Figure 2-2(b) shows a truth table which specifies
the output of the circuit for all possible combinations of
values of the inputs A and B. The first two columns list
the four combinations of values of A and B, and the
next column gives the corresponding values of A' . The

21. ©2004 Brooks/Cole
Figure 2-2: 2-Input Circuit and Truth
Table

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Next, we will use a truth table to specify
the value of Expression (2-1) for all possible
combinations of values of the variables A, B,
and C. On the left side of Table 2-1, we list the
values of the variables A. B. and C. Because each
of the three variables can assume the value 0 or
1, there are 2 x 2 x 2 = 8 combinations of values
of the variables. These combinations are easily
obtained by listing the binary. numbers 000,
001, .. . , 111. In the next three columns of the
truth table, we compute B'. AB'. and AB' + C,
respectively.

23. ©2004 Brooks/Cole
Two expressions are equal if they have the
same value for every possible combination of
the variables. The expression (A + C)(B' +
C) is evaluated using the last three columns of
Table 2-1. Because it has the same value as AB'
+ C for all eight combinations of values of the
variables A, B, and C, we concludeAB' + C = (A + C)(B' + C)
(2-3)
If an expression has n variables, and each
variable can have the value 0 or 1, the number
of different combinations of values of the
variables is

24. ©2004 Brooks/Cole
A B C B’ AB’ AB’+C A+C B’+C (A+C)(B’+C)
0 0 0 1 0 0 0 1 0
0 0 1 1 0 1 1 1 1
0 1 0 0 0 0 0 0 0
0 1 1 0 0 1 1 1 1
1 0 0 1 1 1 1 1 1
1 0 1 1 1 1 1 1 1
1 1 0 0 0 0 1 0 0
1 1 1 0 0 1 1 1 1
Table 2-1: Truth Table for 3
variables
Therefore, a truth table for an n-variable expression
will have 2n rows.

25. ©2004 Brooks/Cole
Section 2.4, p. 38
2.4 Basic Theorems
The following basic laws and theorems of
Boolean algebra involve only a single variable:
Operations with 0 and 1 :
Idempotent laws
Involution law

26. ©2004 Brooks/Cole
Section 2.4, p. 38
Each of these theorems is easily proved by
showing that it is valid for both of the possible
values of X. For example, to prove X +X' = 1,
we observe that if
Laws of
complementarity
Any expression can be substituted for the
variable X in these. theorems. Thus, by
Theorem (2-5),

27. ©2004 Brooks/Cole
Section 2.4, p. 38
We will illustrate some of the basic
theorems with circuits of switches. As before, 0
will represent an open circuit or open switch,
and 1 will represent a closed circuit or closed
switch. If two switches are both labeled with the
variable A, this means that both switches are
open when A = 0 and both are closed when A =
1. Thus the circuit
and by Theorem (2-
8D).

28. ©2004 Brooks/Cole
Section 2.4, p. 38
can be replaced with a
single switch:

29. ©2004 Brooks/Cole
Section 2.4, p. 39
which illustrates the theorem A + A = A. A
switch in parallel with an open circuit is
equivalent to the switch alone

30. ©2004 Brooks/Cole
Section 2.4, p. 39
while a switch in parallel with a short
circuit is equivalent to a short: circuit.

31. ©2004 Brooks/Cole
Section 2.4, p. 39
If a switch is labeled A', then it is open
when A is closed and conversely. Hence, A in
parallel with A' can be replaced with a closed
circuit because one or the other of the two
switches is always closed.

32. ©2004 Brooks/Cole
Section 2.4, p. 39
Similarly, switch A in series with A' can be
replaced with an open circuit (why?).

33. ©2004 Brooks/Cole
Section 2.4, p. 39
Many of the laws of ordinary algebra, such
as the commutative, and associative laws, also
apply to Boolean algebra. The commutative laws
for AND and OR, which follow directly from the
definitions of the AND and OR operations, are
2.5 Commutative, Associative, and
Distributive Laws
This means that the order in which the
variables are written will not affect the result of
applying the AND and OR operations.

34. ©2004 Brooks/Cole
The associative laws also apply to AND and OR:
When forming the AND (or OR) of three
variables, the result is independent of which pair
of variables we associate together first, so
parentheses can be omitted as indicated in
Equations (2-10) and (2-10D).
We will prove the associative law for AND by
using a truth table (Table 2-2). On the left side of the
table, we list all combinations of values of the variables
X. Y, and Z. In the next two columns of the truth table,
we compute XY and YZ for each combination of values
of X. Y and Z. Finally, we compute (XY)Z and X(YZ).
Because (XY)Z and X(17Z) are equal for all possible

35. ©2004 Brooks/Cole
Table 2-2: Proof of Associative Law for
AND

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Figure 2-3 illustrates the associative laws
using AND and OR gates. In Figure 2-3(a)two
two-input AND gates are replaced with a single.
three-input AND gate. Similarly, in Figure 2-3(b)
two two-input OR gates are replaced with a
single three-input OR gate.
Figure 2-3: Associative Law for AND
and OR

37. ©2004 Brooks/Cole
When two or more variables are ANDed
together, the value of the result will be 1 if all of
the variables have the value 1. If any of the
variables have the value 0, the result of the AND
operation will be 0. For example,
When two or more variables are ORed
together, the value, of the result will be 1 if any
of the variables have the value 1. The result of
the OR operation will be o if all of the variables
have the value o. For example,

38. ©2004 Brooks/Cole
Using a truth table, it is easy to show that the
distributive law is valid:
In addition to the ordinary distributive law,
a second distributive law is valid for Boolean
algebra but not for ordinary algebra:

39. ©2004 Brooks/Cole
Proof of the second distributive law follows:
The ordinary distributive law states that the AND
operation distributes over OR, while the second
distributive law states that OR distributes over AND. This
second law is very useful in manipulating Boolean
expressions. In particular, an expression like A + BC,
which cannot be factored in ordinary algebra, is easily
factored using the second distributive, law:

40. ©2004 Brooks/Cole
In each case, one expression can be
replaced by a simpler one. Because each
expression corresponds to a circuit of logic
gates, simplifying an expression leads to
simplifying the corresponding logic circuit
2.6 Simplification Theorems
The following theorems are useful in simplifying
Boolean expressions:

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The proof of the remaining theorems is left as
an exercise.
We will illustrate Theorem (2-14D), using
switches. Consider the following circuit:
Each of the preceding theorems can be
proved by using a truth table, or they can be
proved algebraically starting with the basic
theorems.

42. ©2004 Brooks/Cole
Its transmission is T = Y+ XY' because
there is a closed circuit between the terminals if
switch Y is closed or switch X is closed and
switch Y' is closed. The following circuit is
equivalent because if Y is closed (Y = 1) both
circuits have a transmission of 1; if Y is open
(Y' = 1) both circuits have a transmission of X.Section 2.6, p. 42

43. ©2004 Brooks/Cole
The following example illustrates
simplification of a logic gate circuit using one of
the theorems. In Figure 2-4, the output of circuit (a)
is
Section 2.6, p. 42
By Theorem (2-14), the expression for F
simplifies to AB. Therefore, circuit (a) can be
replaced with the equivalent circuit (b).

44. ©2004 Brooks/Cole
Figure 2-4: Equivalent Gate Circuits
Any expressions can be substituted for X and
Yin the theorems.

45. ©2004 Brooks/Cole
EXAMPLE 1 Simplify Z = A'BC + A'
This expression has the same form as (2-13) if we let X = A' and Y = BC
Therefore, the expression simplifies to Z = X + X Y = X = A'.
Simplify (p. 42-43)

46. ©2004 Brooks/Cole
The following example illustrates
simplification of a logic gate circuit using one of
the theorems. In Figure 2-4, the output of circuit (a)
is
Section 2.6, p. 42
By Theorem (2-14), the expression for F
simplifies to AB. Therefore, circuit (a) can be
replaced with the equivalent circuit (b).

47. ©2004 Brooks/Cole
EXAMPLE 1: Factor A + B'CD. This is of the form X + YZ
where X = A, Y = B', and Z = CD, so
A + B'CD = (X + Y)(X + Z) = (A + B')(A + CD)
A + CD can be factored again using the second distributive law, so
A + B'CD = (A + B')(A + C)(A + D)
EXAMPLE 2: Factor AB' + C'D
EXAMPLE 3: Factor C'D + C'E' + G'H
Factor (p. 44-45)

48. ©2004 Brooks/Cole
Figure 2-5: Circuits for Equations (2-15) and (2-17)

49. ©2004 Brooks/Cole
Figure 2-6: Circuits for Equations (2-18) and (2-20)

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Operations with 0 and 1:
1. X + 0 = X 1D. X • 1 = X
2. X +1 = 1 2D. X • 0 = 0
Idempotent laws:
3. X + X = X 3D. X • X = X
Involution law:
4. (X')' = X
Laws of complementarity:
5. X + X' = 1 5D. X • X' = 0
LAWS AND THEOREMS (a)

51. ©2004 Brooks/Cole
Commutative laws:
6. X + Y = Y + X 6D. XY = YX
Associative laws:
7. (X + Y) + Z = X + (Y + Z) 7D. (XY)Z = X(YZ) = XYZ
= X + Y + Z
Distributive laws:
8. X(Y + Z) = XY + XZ 8D. X + YZ = (X + Y)(X + Z)
Simplification theorems:
9. XY + XY' = X 9D. (X + Y)(X + Y') = X
10. X + XY = X 10D. X(X + Y) = X
11. (X + Y')Y = XY 11D. XY' + Y = X + Y
LAWS AND THEOREMS (b)

52. ©2004 Brooks/Cole
DeMorgan's laws:
12. (X + Y + Z +...)' = X'Y'Z'... 12D. (XYZ...)' = X' + Y' + Z' +...
Duality:
13. (X + Y + Z +...)D = XYZ... 13D. (XYZ...)D = X + Y + Z +...
Theorem for multiplying out and factoring:
14. (X + Y)(X' + Z) = XZ + X'Y 14D. XY + X'Z = (X + Z)(X' + Y)
Consensus theorem:
15. XY + YZ + X'Z = XY + X'Z 15D. (X + Y)(Y + Z)(X' + Z) = (X + Y)(X' + Z)
LAWS AND THEOREMS (c)