Boolean Algebra in Digital Electronics: A Comprehensive Guide with Examples, Study notes of Algebra

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Boolean Algebra

Digital Electronics

What is Boolean Algebra?

Boolean Algebra is a mathematical technique that provides the ability to algebraically simplify logic expressions. These simplified expressions will result in a logic circuit that is equivalent to the original circuit, yet requires fewer gates.

George Boole

George Boole lived in England in the 19th century. His work on mathematical logic , algebra , and the binary number system has had a unique influence upon the development of computers. Boolean Algebra is named after him.

X 0

0

X  0  0

X Y Z

Boolean Theorems (1 of 9)

Single Variable - AND Function

Theorem #

X  X  X

X X

X

Boolean Theorems (3 of 9)

X Y Z

0 0 0

0 1 0

1 0 0

1 1 1

Single Variable - AND Function

Theorem #

X (^0)

X

X  X  0

Boolean Theorems (4 of 9)

X Y Z

0 0 0

0 1 0

1 0 0

1 1 1

Single Variable - AND Function

Theorem #

X  1  1

Boolean Theorems (6 of 9)

X Y Z

X 1

1

Single Variable - OR Function

Theorem #

X  X  X

Boolean Theorems (7 of 9)

X Y Z

X X

X

Single Variable - OR Function

Theorem #

X  X 0 1 0

1 0 1

Single Variable - Invert Function

Boolean Theorems (9 of 9)

X

X

X

X X X

Theorem #

Example #1: Boolean Algebra

Simplify the following Boolean expression and note the Boolean theorem used at each step. Put the answer in SOP form.

F 1  AABCCD

Step #1: Boolean Algebra

F 1  A A B  C CD

Which Theorem can be applied to AAB?

Theorem # 3:

Step #1: Boolean Algebra

F 1  A A B  C CD

Which Theorem can be applied to AAB?

Theorem # 3: X  X  X

Step #1: Boolean Algebra

F 1  A A B  C CD

Which Theorem can be applied to AAB?

Theorem # 3: X  X  X

AAB can be simplified to AB

F 1  A B  CCD

Step #2: Boolean Algebra

F 1  A B  CCD