Digital Electronic Systems Unit 6

Rules and laws of Boolean Algebra

Laws of Boolean AlgebraThe three basic laws of Boolean Algebra are:

(1) Commutative Laws

(a) Law of Addition

(b) Law of Multiplication

(2) Associative Laws

(a) Law of Addition

(b) Law of Multiplication

(3) Distributive Law

Commutative LawsThe commutative law of Boolean addition for two variables is written as:

The law states that the order in which the variables are ORed makes no difference to the outcome. This law can be extended to any number of variables.

A B A+B0 0 00 1 11 0 11 1 1

B A B+A0 0 00 1 11 0 11 1 1

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Digital Electronic Systems Unit 6

The commutative law of Boolean Multiplication for two variables is written as:

The law states that the order in which the variables are ANDed makes no difference to the outcome. This law can be extended to any number of variables.

A B A.B0 0 00 1 01 0 01 1 1

B A B.A0 0 00 1 01 0 01 1 1

Associative LawsThe associative law of Boolean addition for three variables is written as:

The law states that the result of ORing several variables together is the same regardless of the grouping of these variables. This law can be extended to any number of variables.

A B C A+B (A+B)+C0 0 0 0 00 0 1 0 10 1 0 1 10 1 1 1 11 0 0 1 11 0 1 1 11 1 0 1 11 1 1 1 1

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A B C B+C A+(B+C)0 0 0 0 00 0 1 1 10 1 0 1 10 1 1 1 11 0 0 0 11 0 1 1 11 1 0 1 11 1 1 1 1

The associative law of Boolean multiplication for three variables is written as:

The law states that the result of ANDing several variables together is the same regardless of the grouping of these variables. This law can be extended to any number of variables.

A B C A.B (A.B).C0 0 0 0 00 0 1 0 00 1 0 0 00 1 1 0 01 0 0 0 01 0 1 0 01 1 0 1 01 1 1 1 1

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Digital Electronic Systems Unit 6

A B C B.C A.(B.C)0 0 0 0 00 0 1 0 00 1 0 0 00 1 1 1 01 0 0 0 01 0 1 0 01 1 0 0 01 1 1 1 1

Distributive LawThe distributive law for three variables is written as:

The law states that ORing two or more variables and then ANDing the result with a single variable is the same as ANDing the single variable with each of the two or more variables and then ORing the result. This law can be extended to any number of variables.

A B C B+C A.(B+C)0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 1 01 0 0 0 01 0 1 1 11 1 0 1 11 1 1 1 1

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Digital Electronic Systems Unit 6

A B C A.B A.C A.B+A.C0 0 0 0 0 00 0 1 0 0 00 1 0 0 0 00 1 1 0 0 01 0 0 0 0 01 0 1 0 1 11 1 0 1 0 11 1 1 1 1 1

Rules of Boolean Algebra

(1) (7)(2) (8)(3) (9)(4) (10)(5) (11)(6) (12)

These 12 rules are useful in manipulating and simplifying Boolean expressions.

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Digital Electronic Systems Unit 6

Rule 1

This rule states that if you OR a variable with 0 then you will get that variable at the output. Shown below is a diagram describing this rule.

INPUT A INPUT B OUTPUT0 0 00 1 11 0 11 1 1

Rule 2

This rule states that if you OR a variable with 1 then you will always get a 1 (logic HIGH) at the output. Shown below is a diagram describing this rule.

INPUT A INPUT B OUTPUT0 0 00 1 11 0 11 1 1

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Digital Electronic Systems Unit 6

Rule 3

When a 0 is ANDed with any logic level then the output of the AND gate is always 0 (logic LOW). Shown below is a diagram describing the rule.

INPUT A INPUT B OUTPUT0 0 00 1 01 0 01 1 1

Rule 4

When a 1 is ANDed with any variable logic level then the output of the AND gate is always that variable logic level. Shown below is a diagram describing the rule.

INPUT A INPUT B OUTPUT0 0 00 1 01 0 01 1 1

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Digital Electronic Systems Unit 6

Rule 5

This rule states that the output of an OR gate when the two inputs are the same is the same as the inputs. Shown below is a diagram describing the rule.

INPUT A INPUT B OUTPUT0 0 00 1 11 0 11 1 1

Rule 6

This rule states that when a variable is ORed with the variables inverse, the answer is the always 1 (logic HIGH). Shown below is a diagram describing the rule.

INPUT A INPUT B OUTPUT0 0 00 1 11 0 11 1 1

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Digital Electronic Systems Unit 6

Rule 7

This rule states that if a variable is ANDed with itself then the output of the AND gate is the variable input itself. Shown below is a diagram describing the rule.

INPUT A INPUT B OUTPUT0 0 00 1 01 0 01 1 1

Rule 8

This rule states that when a variable is ANDed with its inverse then the output of the AND gate is always 0. Shown below is a diagram describing the rule.

INPUT A INPUT B OUTPUT0 0 00 1 01 0 01 1 1

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Digital Electronic Systems Unit 6

Rule 9

This rule states that any variable that goes through two NOT gates successively will be returned to its original logic level. Shown below is a diagram describing the rule.

Rule 10

This rule uses the Distributive Law in its proof. The proof is as follows:

Shown below is a diagram describing the rule.

INPUTA

INPUTB

A.B OUTPUTA + A.B

0 0 0 00 1 0 01 0 0 11 1 1 1

IDENTICAL

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Rule 11

Rule 11 is proved as follows

Shown below is a diagram describing the rule.

The proof can also be derived using the truth table shown below

INPUTA

INPUTB

OUTPUTS A + B

0 0 0 0 00 1 1 1 11 0 0 1 11 1 0 1 1

IDENTICAL

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Rule 12

This rule is proved as follows( A + B ) . ( A + C ) = A . A + A . C + A . B + B . C Distributive Law

= A + A . C + A . B + B . C Rule 7 = A .( 1 + C ) + A . B + B . C Distributive Law = A . 1 + A . B + B . C Rule 2 = A .( 1 + B ) + B . C Distributive Law = A . 1 + B . C Rule 2 = A + B . C Rule 4

Shown below is a diagram describing the rule.

The proof can also be derived using the truth table shown below

INPUTA

INPUTB

INPUTC

A+B A+C B.C OUTPUT OUTPUT(A+B).(A+C) A+B.C

0 0 0 0 0 0 0 00 0 1 0 1 0 0 00 1 0 1 0 0 0 00 1 1 1 1 1 1 11 0 0 1 1 0 1 11 0 1 1 1 0 1 11 1 0 1 1 0 1 11 1 1 1 1 1 1 1

IDENTICAL

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Digital Electronic Systems Unit 6

De Morgan’s Theorems

De Morgan’s Theorems are expressed for two variables in the following way.

Equation 1Equation 2

Equation 1 states that the complement of two or more variables ANDed together is equivalent to the OR of the complements of the individual variables. This can be seen very clearly in the diagram and truth table below.

Identical

Equation 2 states that the complement of two or more variables ORed together is equivalent to the AND of the complements of the individual variables. This can be more clearly seen in the diagram and the truth table below.

X Y X+Y 0 0 1 1 0 1 10 1 1 0 1 0 01 0 0 1 1 0 01 1 0 0 1 0 0

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X Y X.Y

0 0 1 1 0 1 10 1 1 0 0 1 11 0 0 1 0 1 11 1 0 0 1 0 0

Digital Electronic Systems Unit 6

IdenticalDe Morgan’s Theorems are not specific to just two variables. They can be applied to more than two variables. For three variables, De Morgan’s theorems are written as follows.

Terminology: In Boolean algebra, a product term is defined as a term consisting of variables (or their complements) that are ANDed together. When two or more product terms are ORed together, the resultant expression is in sum of products (SOP) format.

Ex. 1: Apply De Morgan’s Theorems to reduce the expression to sum of products (SOP) format.

Ex. 2: Apply De Morgan’s Theorems to reduce the expression to sum of products (SOP) format.

Ex. 3: Apply De Morgan’s Theorems to reduce the expression to sum of products (SOP) format.

Ex. 4: Apply De Morgan’s Theorems to reduce the expression to sum of products (SOP) format.

Ex. 5: Apply De Morgan’s Theorems to reduce the expression to sum of products (SOP) format.

Ex. 6: Apply De Morgan’s Theorems to reduce the expression to sum of products (SOP) format.

Ex. 7: Apply De Morgan’s Theorems to reduce the expression to sum of products (SOP) format.

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