Boolean functions

Boolean expressions and Boolean functions

Boolean expressions and Boolean functionsии

Boolean expressions and Boolean functions

Example 5

Boolean expressions and Boolean functions

The Boolean functions of degree two

Boolean expressions and Boolean functions

Identities of Boolean algebra

Identities of Boolean algebra
Compare these Boolean identities with the logical
equivalences and the

set identities!

Identities of Boolean algebra

Disjunctive normal form
We now show how any Boolean expression can be expressed

in an equivalent standard form (called the disjunctive normal form).

Disjunctive normal form

Example of a minterm

Disjunctive normal form

A procedure for constructing a Boolean expression representing a function with given

values as DNF

By taking Boolean sums of distinct minterms we can build up a Boolean expression with a specified set of values.
In particular, a Boolean sum of minterms has the value 1 when exactly one of the minterms in the sum has the value 1.
It has the value 0 for all other combinations of values of the variables.

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A procedure for constructing a Boolean expression representing a function with given

values as DNF

Consequently, given a Boolean function, a Boolean sum of minterms can be formed that has the value 1 when this Boolean function has the value 1, and has the value 0 when the function has the value 0.
The minterms in this Boolean sum correspond to those combinations of values for which the function has the value 1.

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Disjunctive normal form

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Solution of example 9

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Conjunctive normal form

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Example of a maxterm

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Conjunctive normal form

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A procedure for constructing a Boolean expression representing a function with given

values as CNF

By taking Boolean product of distinct maxterms we can build up a Boolean expression with a specified set of values.
In particular, a Boolean product of maxterms has the value 0 when exactly one of the maxterms in the product has the value 0.
It has the value 1 for all other combinations of values of the variables.

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A procedure for constructing a Boolean expression representing a function with given

values as CNF

Consequently, given a Boolean function, a Boolean product of maxterms can be formed that has the value 0 when this Boolean function has the value 0, and has the value 1 when the function has the value 1.
The maxterms in this Boolean sum correspond to those combinations of values for which the function has the value 0.

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Conjunctive normal form

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Solution of example 10

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Functional completeness

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Functional completeness
Can we find a smaller set of functionally complete operators?
We

can do so if one of the three operators of this set can be expressed in terms of the other two.

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Functional completeness

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Functional completeness

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Functional completeness
We have found sets containing two operators that are functionally complete.

Can we find a smaller set of functionally complete operators, namely, a set containing just one operator? Such sets exist.

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Functional completeness

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Functional completeness

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Functional completeness

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Biography
George Boole, the son of a cobbler, was born in Lincoln, England,

in November 1815. Because of his family’s difficult financial situation, Boole struggled to educate himself while supporting his family.
Nevertheless, he became one of the most important mathematicians of the 1800s. Although he considered a career as a clergyman, he decided instead to go into teaching, and soon afterward opened a school of his own.

George Boole
1815 – 1864

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Biography
In his preparation for teaching mathematics, Boole – unsatisfied with textbooks of

his day – decided to read the works of the great mathematicians.
While reading papers of the great French mathematician Lagrange, Boole made discoveries in the calculus of variations, the branch of analysis dealing with finding curves and surfaces by optimizing certain parameters.

George Boole
1815 – 1864

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Biography
In 1848 Boole published The Mathematical Analysis of Logic, the first of

his contributions to symbolic logic.
In 1849 he was appointed professor of mathematics at Queen’s College in Cork, Ireland.
In 1854 he published The Laws of Thought, his most famous work. In this book, Boole introduced what is now called Boolean algebra in his honor.

George Boole
1815 – 1864