"Support of teaching technical subjects in English “

1. "Support ofteachingtechnicalsubjects in English“ Learning program: Mechanic – electrician Name of the program: Numerical systems II. class Two-state 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.

2. Numerical systems

3. Two-state logic, Boolean algebra 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 statescan 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 understooda logical operation. Basic operations used in numerical method are logical addition (A + B), logical product (A . B) and negation ( A ) [read non A]. -

4. Boolean algebra 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

5. Conversion of integers from the decimal numerical system to the binary system 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) - - - - - -

6. Conversion of decimal integers to binary 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.

7. Conversion of decimal numbersto binary The logical product is represented by the conjunction AND. For the logical product this veracity chart holds true: 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. 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.

8. Activity for pupils - Game The End Question chart: for100 for 300 for500 1 1 1 Prémie Prémie 2 2 2 3 3 3 Prémie A B C D E F G H

9. Activity for pupils - Game Question for 100 How many different states can the two-state logic have?

10. Activity for pupils - Game Question for 100 Which century did George Boole live in?

11. Activity for pupils - Game Question for 100 Where did George Boole live?

12. Activity for pupils - Game Question for 300 What is the commutative law?

13. Activity for pupils - Game Question for 300 What is the associative law?

14. Activity for pupils - Game Question for 300 Which logical conjunction represents the logical addition?

15. Activity for pupils - Game Question for 500 Which logical conjunction represents the logical product?

16. Activity for pupils - Game Question for 500 Which century did George Boole live in?

17. Activity for pupils - Game Question for 500 What is enabled by the logical function of one variable negation?

18. Literature • 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