Section 3.4

Boolean Algebra

Boolean Algebra

A link between:Section 1.3: Logic SystemsSection 3.3: Set Systems

Application:Section 3.5: Logic Circuits in Computer

Science

Recall:

We have already studied two systems: logic and sets, and have observed several properties that each system possesses.

Theorem 2, Section 1.3: Let p, q, r be propositions, and let t indicate a tautology and c a contradiction. The logical

equivalences shown below hold:

Theorem 6, Section 3.3: For sets A, B, and C, the universal set U and the empty set, the following properties hold:

Similarity between the theorems:

Change p,q,r to A, B, C Change to Change to Change to Change to = Change t to U Change c to {}

'

Practice: Convert the logical expression to set theory notation, using sets A,B, and C:

)()( prqp

Practice: Convert the set theory expression to logical notation, using logical variables, p, q, and r:

)(')'( BAABA

Practice: Verify by quoting logic properties

that =_______________________ by

=_______________________by

=_______________________by

=_______________________by

=_______________________by

=

pqqpq )(

qpq )(

pq

Introduction to Boolean Algebra

In the mid-1800’s, the English mathematician George Boole investigated systems having properties like those shared by sets and logic systems.

We will use the following notation when describing a Boolean algebra: lowercase letters and + for the operations 0 and 1 for special symbols

Connections between Logic, Sets, and Boolean Algebra

Properties of a Boolean AlgebraCompare this to the properties for sets and logic.

Any logical expression or expression of set theory can be written using Boolean algebra notation.

Write the following using Boolean algebra notation, with variables a and b:

1)

2)

)'()( ABAABA

tqpqp )()(

Verify the following Boolean algebra equality by quoting properties of a Boolean algebra:

abbaa )(

Verify the following Boolean algebra equality by quoting properties of a Boolean algebra:

bcabcab )(

Advantages of the Boolean algebra system:

Some properties are analogous to familiar properties in algebra, e.g. the distributive, commutative, and associative properties.

Symbolic manipulation is easier with a Boolean system than with a logic or set system.

Duality

The dual of a Boolean algebra expression is obtained by interchanging the roles of and + , and also interchanging the roles of 0 and1.

Example: The dual of is

Theorem: For every true equality in a Boolean algebra, the “dual” of that property is also true.

)1)(( cba 0 cab