Department of electronics and radio engineering т basic laws of Boolean algebra

DEPARTMENT OF ELECTRONICS AND RADIO ENGINEERING

Т 4. Basic laws of Boolean algebra

Plan

• Number systems
• Logic functions
• Boolean Algebra Laws
• Logical operations

Number systems

• Numbering is a way of expressing and denoting numbers.
• An arbitrary number Q in any positional number system with base q can be represented as a polynomial:

;

• where хi – bit coefficient (хi=0…q-1);
• qi – weight coefficient.

Number systems

• From the point of view of electronics, the criterion for choosing q is to minimize hardware costs while providing sufficient noise immunity. Preference was given to the binary system, since electronic devices must have only two stable states. Then, in order to distinguish signals in this system, it is enough to answer the question: is there an impulse or not?

Number systems

• For example, a decimal number Х=29 in the binary system will be represented as:

 

29 = 1·24 + 1·23 + 1·22 + 0·21 + 1·20,

• and symbolically - a sequence of numbers 11101.

Number systems

• Thus, in the binary number system, any number can be written using the digits 0 and 1. To represent these numbers in digital systems, it is enough to have electronic circuits that can be in two states with clearly different values ​​of any electrical quantity (potential or current). One of the values ​​of this quantity is "assigned" the number 0, the other 1.

Logic functions

• Logical expressions х1, х2, х3, …хn are functions of variables with a value of 0 or 1.
• If the number of logical variables is n, then using 1 and 0 you can get 2n combinations. Example,
• If n = 1: then x = 0 and x = 1;
• If n = 2: then x1x2 = 00,01,10,11.
• For each variable y can be 0 or 1.

Logic variables

• Each statement corresponds to only one of two meanings: either "true" or "false", which are logical constants. Boolean variables can only take one of two possible values: "true" or "false".
• Logical variables can take one of two constant values:

x = 0, if x ¹ 1; x = 1 if x ¹ 0.

• We can perform logical operations on logical constants and variables: logical addition, logical multiplication and negation (inversion).

Logic functions

• logical negation (inversion, NOT procedure), is performed with the "-" sign over the variable;
• Logical addition (disjunction, OR), “+”;
• Logical multiplication (conjunction, AND), “·”.

Boolean laws

Boolean laws

Boolean laws

Control questions

• Number systems
• Logic functions
• Boolean Algebra Laws
• Logical operations