01 boolean algebra

EBB4453 Digital System Design

Boolean Algebra

A quick review

Boolean Algebra 2

Learning Outcomes

At the end of the lecture, students are able to:

Recall the 3 basic laws of Boolean Algebra:

a) Commutative law

b) Associative law

c) Distributive law

Recall the 12 rules of Boolean algebra.

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EBB4453 Boolean Algebra 3

Boolean Algebra

Mathematics of digital systems.

There are 3 basic laws and 12 basic rules in

Boolean algebra.

It is very important for you to understand and

remember these rules and laws as they are

very useful in simplifying your digital circuits.

Boolean Algebra 4

Boolean Addition and Multiplication

The rules for Boolean Addition are:

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 1

0 0 = 0

0 1 = 0

1 0 = 0

1 1 = 1

The rules for Boolean Multiplication are:

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Boolean Algebra 5

Laws of Boolean Algebra

The three basic laws of Boolean algebra are:

Commutative law

Associative law

Distributive law

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Boolean Algebra 6

A

B A B

B

A BA

A

B A + B

B

A B + A

Commutative Law

The order in which the variables are ORed or

ANDed does NOT matter.

Commutative law for addition(OR):

A + B = B + A

AB = BA

Commutative law for multiplication(AND):

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Boolean Algebra 7

A

B

C

A + (B + C) A

B

C

(A + B) + C

A

B

C

A (BC) A

B

C

(A B)C

The order the variables are grouped when ANDing or ORing more than 2 variables does NOT matter.

Associative law for addition(OR):

Associative Law

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

A(BC) = (AB)C

Associative law for multiplication(AND):

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Boolean Algebra 8

Distributive Law

ORing 2 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.

A(B + C) = AB + AC

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Rules of Boolean Algebra

Rule 1: A + 0 = A

A variable ORed with 0 is always equal to the

variable

Proof : When A = 0

When A = 1

Rule 2: A + 1 = 1

A variable ORed with 1 is always equal to 1

Proof: When A = 0

When A = 1

0 + 0 = 0 = A

1 + 0 = 1 = A

0 + 1 = 1

1 + 1 = 1

Boolean Algebra 10


Rule 3: A 0 = 0

A variable ANDed with 0 is always equal to 0

Proof : When A = 0

When A = 1

0 0 = 0

Rule 4: A 1 = A

A variable ANDed with 1 is always equal to the

variable

Proof: When A = 0

When A = 1

1 0 = 0

0 1 = 0 = A

1 1 = 1 = A

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Boolean Algebra 11


Rule 5: A + A = A

A variable ORed with itself is always equal to the

variable

Proof : When A = 0

When A = 1

Rule 6: A + A’ = 1

A variable ORed with its complement is always equal to 1

Proof: When A = 0

When A = 1

0 + 0 = 0 = A

1 + 1 = 1 = A

1 + 0 = 1

0 + 1 = 1

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Boolean Algebra 12


Rule 7: A A = A

A variable ANDed with itself is always equal to the

variable

Proof : When A = 0

When A = 1

Rule 8: A A’ = 0 A variable ANDed with its complement is always

equal to 0

Proof: When A = 0

When A = 1

0 0 = 0 = A

1 1 = 1 = A

0 1 = 0

1 0 = 0

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Rule 9: A’’ = A

The double complement is always equal to the

variable

Proof :

Rule 10: A + AB = A

Proof: A(1+B) Distributive law

1+B = 1 Rule 2

A(1) = A Rule 4

0 0 1

1 1 0

Boolean Algebra 14


Rule 11: A + A’B= A + B

Proof : A = A + AB Rule 10

A + A’B = A + AB + A’B

= A + B(A + A’) Distributive Law

= A + B(1) Rule 6

= A + B Rule 4

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Rule 12: (A + B)(A + C) = A + BC

Proof: (A + B) (A +C)

= AA +AC + AB + BC expand

= A + AC + AB +BC Rule 7

= A(1 + C + B) + BC factorize

= A(1) + BC Rule 2

= A + BC Rule 4

Boolean Algebra 16

Summary of Boolean Algebra

Commutative laws : A+B = B + A

AB = BA

Associative laws: A+(B+C) = (A+B)+C

A(BC) = (AB)C

Distributive laws: A(B+C) = AB + AC

A + 0 = A

A + 1 = 1

A 0 = 0

A 1 = A

A + A = A

A + A’ = 1

A A = A

A A’ = 0

A’’ = A

A + AB = A

A + A’B = A + B

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

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Boolean Algebra 17

Homework

Proof (to yourself) rule 10 to rule 12.

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01 boolean algebra

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