4 BOOLEAN ALGEBRA
AND
LOGIC SIMPLIFICATION
BOOLEAN OPERATIONS AND EXPRESSIONS
Variable, complement, and literal are terms used in Boolean algebra. A
variable is a symbol used to represent a logical quantity. Any single variable
can have a 1 or a 0 value. The complement is the inverse of a variable and is
indicated by a bar over variable (overbar). For example, the complement of
the variable A is A. If A = 1, then A = 0. If A = 0, then A = 1. The
complement of the variable A is read as "not A" or "A bar." Sometimes a
prime symbol rather than an overbar is used to denote the complement of a
variable; for example, B' indicates the complement of B. A literal is a
variable or the complement of a variable.
Boolean Addition
Recall from part 3 that Boolean addition is equivalent to the OR
operation. In Boolean algebra, a sum term is a sum of literals. In logic
circuits, a sum term is produced by an OR operation with no AND
operations involved. Some examples of sum terms are A + B, A + B, A +
B + C, and A + B + C + D.
A sum term is equal to 1 when one or more of the literals in the term are 1. A
sum term is equal to 0 only if each of the literals is 0.
Example
Determine the values of A, B, C, and D that make the sum term
A+B+C+D equal to 0.
,Boolean Multiplication
Also recall from part 3 that Boolean multiplication is equivalent to the AND
operation. In Boolean algebra, a product term is the product of literals. In
logic circuits, a product term is produced by an AND operation with no OR
operations involved. Some examples of product terms are AB, AB, ABC,
and ABCD.
A product term is equal to 1 only if each of the literals in the term is 1. A
product term is equal to 0 when one or more of the literals are 0.
Example
Determine the values of A, B, C, and D that make the product term ABCD
equal to 1.
LAWS AND RULES OF BOOLEAN ALGEBRA
■ Laws of Boolean Algebra
The basic laws of Boolean algebra-the commutative laws for addition and
multiplication, the associative laws for addition and multiplication, and the
distributive law-are the same as in ordinary algebra.
Commutative Laws
►The commutative law of addition for two variables is written as
A+B = B+A
This law states that the order in which the variables are ORed makes no
difference. Remember, in Boolean algebra as applied to logic circuits,
addition and the OR operation are the same. Fig.(4-1) illustrates the
commutative law as applied to the OR gate and shows that it doesn't matter
to which input each variable is applied. (The symbol ≡ means "equivalent
to.").
, Fig.(4-1) Application of commutative law of addition.
►The commutative law of multiplication for two variables is
A.B = B.A
This law states that the order in which the variables are ANDed makes no
difference. Fig.(4-2), il1ustrates this law as applied to the AND gate.
Fig.(4-2) Application of commutative law of multiplication.
Associative Laws :
►The associative law of addition is written as follows for three variables:
A + (B + C) = (A + B) + C
This law states that when ORing more than two variables, the result is the
same regardless of the grouping of the variables. Fig.(4-3), illustrates this
law as applied to 2-input OR gates.
Fig.(4-3) Application of associative law of addition.
►The associative law of multiplication is written as follows for three
variables:
A(BC) = (AB)C
, This law states that it makes no difference in what order the variables are
grouped when ANDing more than two variables. Fig.(4-4) illustrates this law
as applied to 2-input AND gates.
Fig.(4-4) Application of associative law of multiplication.
Distributive Law:
►The distributive law is written for three variables as follows:
A(B + C) = AB + AC
This law states that ORing two or more variables and then ANDing the result
with a single variable is equivalent to ANDing the single variable with each
of the two or more variables and then ORing the products. The distributive
law also expresses the process of factoring in which the common variable A
is factored out of the product terms, for example,
AB + AC = A(B + C).
Fig.(4-5) illustrates the distributive law in terms of gate
implementation.
Fig.(4-5) Application of distributive law.
AND
LOGIC SIMPLIFICATION
BOOLEAN OPERATIONS AND EXPRESSIONS
Variable, complement, and literal are terms used in Boolean algebra. A
variable is a symbol used to represent a logical quantity. Any single variable
can have a 1 or a 0 value. The complement is the inverse of a variable and is
indicated by a bar over variable (overbar). For example, the complement of
the variable A is A. If A = 1, then A = 0. If A = 0, then A = 1. The
complement of the variable A is read as "not A" or "A bar." Sometimes a
prime symbol rather than an overbar is used to denote the complement of a
variable; for example, B' indicates the complement of B. A literal is a
variable or the complement of a variable.
Boolean Addition
Recall from part 3 that Boolean addition is equivalent to the OR
operation. In Boolean algebra, a sum term is a sum of literals. In logic
circuits, a sum term is produced by an OR operation with no AND
operations involved. Some examples of sum terms are A + B, A + B, A +
B + C, and A + B + C + D.
A sum term is equal to 1 when one or more of the literals in the term are 1. A
sum term is equal to 0 only if each of the literals is 0.
Example
Determine the values of A, B, C, and D that make the sum term
A+B+C+D equal to 0.
,Boolean Multiplication
Also recall from part 3 that Boolean multiplication is equivalent to the AND
operation. In Boolean algebra, a product term is the product of literals. In
logic circuits, a product term is produced by an AND operation with no OR
operations involved. Some examples of product terms are AB, AB, ABC,
and ABCD.
A product term is equal to 1 only if each of the literals in the term is 1. A
product term is equal to 0 when one or more of the literals are 0.
Example
Determine the values of A, B, C, and D that make the product term ABCD
equal to 1.
LAWS AND RULES OF BOOLEAN ALGEBRA
■ Laws of Boolean Algebra
The basic laws of Boolean algebra-the commutative laws for addition and
multiplication, the associative laws for addition and multiplication, and the
distributive law-are the same as in ordinary algebra.
Commutative Laws
►The commutative law of addition for two variables is written as
A+B = B+A
This law states that the order in which the variables are ORed makes no
difference. Remember, in Boolean algebra as applied to logic circuits,
addition and the OR operation are the same. Fig.(4-1) illustrates the
commutative law as applied to the OR gate and shows that it doesn't matter
to which input each variable is applied. (The symbol ≡ means "equivalent
to.").
, Fig.(4-1) Application of commutative law of addition.
►The commutative law of multiplication for two variables is
A.B = B.A
This law states that the order in which the variables are ANDed makes no
difference. Fig.(4-2), il1ustrates this law as applied to the AND gate.
Fig.(4-2) Application of commutative law of multiplication.
Associative Laws :
►The associative law of addition is written as follows for three variables:
A + (B + C) = (A + B) + C
This law states that when ORing more than two variables, the result is the
same regardless of the grouping of the variables. Fig.(4-3), illustrates this
law as applied to 2-input OR gates.
Fig.(4-3) Application of associative law of addition.
►The associative law of multiplication is written as follows for three
variables:
A(BC) = (AB)C
, This law states that it makes no difference in what order the variables are
grouped when ANDing more than two variables. Fig.(4-4) illustrates this law
as applied to 2-input AND gates.
Fig.(4-4) Application of associative law of multiplication.
Distributive Law:
►The distributive law is written for three variables as follows:
A(B + C) = AB + AC
This law states that ORing two or more variables and then ANDing the result
with a single variable is equivalent to ANDing the single variable with each
of the two or more variables and then ORing the products. The distributive
law also expresses the process of factoring in which the common variable A
is factored out of the product terms, for example,
AB + AC = A(B + C).
Fig.(4-5) illustrates the distributive law in terms of gate
implementation.
Fig.(4-5) Application of distributive law.
