Learning program: Mechanic – electrician

Name of the program: Numerical systems II. class Two-staTwo-statte logic, Boolean algebrae logic, Boolean algebra

Made by: Mgr. Holman Pavel

Projekt Anglicky v odborných předmětech, CZ.1.07/1.3.09/04.0002

je spolufinancován Evropským sociálním fondem a státním rozpočtem

České republiky.

Human and machine decision-making have the similarity that they both take place on the level of simplest decision-making operations. These decision-making situations have only two possible conclusions – YES or NO.In digital systems YES is substituted with log.1 and NO with log. 0. This logic where only two possible states can exist is called two-stage logic.

Logical function As well as humans a digital system makes decisions based on input information. For example, if we want to cross the street, we keep eye on these information: there is/isn‘t a car, there is/isn‘t the crossing, there is/isn‘t traffic lights, there is/isn‘t the green light etc. In the numerical method we call the input information logical variable. Logical variables are transformed to inputs of the digital system. Decision-making is understood a logical operation. Basic operations used in numerical method are logical addition (A + B), logical product (A . B) and negation ( A ) [read non A].-

This algebra was formed by British mathematician George S. Boole (1815 – 1864) as an aid for illustration of philosophical problems using mathematic apparatus based on two verity values. Boole‘s work was nearly forgotten. Almost 100 years ago it was discovered and used by mathematician Claude E. Shannon, who pointed out capability of Boolean algebra to describe attributes and design of relay circuits. Boolean algebra is widely used in particular in design of logical circuits made of signal boxes, that means product elements, addition elements and invertors, which can be used for direct realization of basic operations of Boolean algebra. George Boole

Rules of Boolean algebra enable to operate not only with logical variables, but even with entire functions. While using these rules we work with basic logical operations and their attributes. Rules of the Boolean algebra are presented in the following chart.

For arbitrary elements a; b; c of the Boolean algebra B stands:(1) a + a = a,(2) a + b = b + a (commutativity),(3) a + (b + c) = (a + b) + c (associativity),(4) a + (ab) = a,(5) a(b + c) = (ab) + (ac) (distributivity),(6) a + 0 = a,(7) a * 0 = 0,(8) 1 = 0,(9) a + a = 1,(10) a = a,(11) a + b = a * b (De Morgan‘s law),and also dual forms of all these statements (in which we switch symbols + and * and symbols 0 and 1)

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The logical addition is represented by logical conjunction OR If we have two logical variances, in case of OR this veracity chart holds true:

As you can see in the chart, same symbol is usually used for logical addition and for algebraic addition. Sometimes there are different symbols for that (for example A ^ B etc.). The chart shows us that for the result of the logical addition be equal 1 it is sufficient that at least one of the logical variances attain the value of 1.

 A B A + B

0 0 0

0 1 1

1 0 1

1 1 1

The logical product is represented by the conjunction AND.

For the logical product this veracity chart holds true:

Except of these two operations for logical functions of two variables it is necessary to implement the logical function of one variable. It is useful to implement function of negation, which means function assigning to the logical variable that value, which it doesn‘t have, for example for the logical null assigns one and the other way around.

 A B A . B

0 0 0

0 1 0

1 0 0

1 1 1

For the logical product we use symbols A.B alternatively AB; other symbols like for example A Ú B are used rarely. To attain the value 1 of the logical product result it is necessary that all logical variances constituting this product attain the value 1, or the result will attain the value 0.

Question chart:

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for 100 for 500 for 300

A B C D

E F G H

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Question for 100

How many different states can the two-state logic have?

Question for 100

Which century did George Boole live in?

Question for 100

Where did George Boole live?

Question for 300

What is the commutative law?

Question for 300

What is the associative law?

Question for 300

Which logical conjunction represents the logical addition?

Question for 500

Which logical conjunction represents the logical product?

Question for 500

Which century did George Boole live in?

Question for 500

What is enabled by the logical function of one variable negation?

Mužík, J. Management ve vzdělávání dospělých. Praha: EUROLEX BOHEMIA, 2000. ISBN 80-7361-269-7.

Operační program Vzdělávání pro konkurenceschopnost, ESF 2007 – 2013. Dostupné na: http://www.msmt.cz/eu/provadeci-dokument-k-op-vzdelavani-pro-

konkurenceschopnost MALINA, V. Digitální technika. České Budějovice: KOPP, 1996 KRÝDL, M. Číslicová technika. Dubno, 1999 PODLEŠÁK, J., SKALICKÝ, P. Spínací a číslicová technika. Praha, 1994 PECINA, J. Ing. PaedDr. CSc.; PECINA, P. Mgr. Ph.d. Základy císlicové techniky. Brno,

2007