Topic: Logical operations and logical elements.
Goal: to form concepts in students: logical quantities, logical operations, to teach how to build truth tables, to form an understanding of the devices of the computer element base
Method: story, conversation, problem solving
Technology: person-centered
Teaching software: PC, information sheet
During the classes:
1. Organizing time.
- greeting students
- checking readiness for the lesson.
2. Setting lesson goals:
- how does a person think? What science studies the forms and methods of human thinking?
- arithmetic multiplication and logical multiplication. What are the similarities and differences?
- what is an inference?
3. Presentation of new material
Logical operations
A logical variable is a simple statement containing only one thought.
Its symbolic designation is a Latin letter (A, B, X, Y,...). The value of a Boolean variable can only be the constant TRUE or FALSE. (1 and 0).
Composite statement - logical a function that contains several simple thoughts connected to each other using logical operations.
Logical operations - logical action.
LOGICAL OPERATIONS
Conjunction Disjunction Inversion
Conjunction
(from Latin - I connect) Disjunction
(I distinguish from Latin)
Inversion
(from lat. - turn over)
Name Logical multiplication Logical addition negation
Designation A&B, A B A v B, A+B A, Ā
Conjunction in natural language A and B A or B Not A
Examples:
A = “The number is 10-even”
B= “The number 10 is negative” “The number 10 is even and negative” - FALSE “The number 10 is even or negative” - TRUE “It is not true that the number 10 is even” - FALSE
“It is not true that the number 10 is negative” - TRUE
Truth table A B A& B A B A v B A A
0 0 0 0 0 0 0 1
0 1 0 0 1 1
1 0 0 1 0 1 1 0
1 1 1 1 1 1
A truth table is a table that shows by actions what values a logical expression takes for all possible sets of its variables.
ALGORITHM FOR COMPILATION OF TRUTH TABLES:
1. Find out the number of rows in the table (2n,n is the number of variables)
2. Find out the number of columns = number of variables + number of logical operations
3. Establish the sequence of logical operations
4. Construct a table, indicating the names of the columns and possible sets of values of the original logical variables.
5. Fill out the truth table by column.
Task: Create a truth table for the expression F=(A v B)& (A v B)
A B A v B A B A v B (A v B)& (A v B)
0 0 0 1 1 1 0
0 1 1 1 0 1 1
1 0 1 0 1 1 1
1 1 1 0 0 0 0
Logic elements
Elements that implement basic logical operations are called basic logical elements or gates, and they are characterized not by the state of the contacts, but by the presence of signals at the input and output of the element.
Logic elements
CONJUNCTION DISJUNCTION INVERSION
conjunctor disjunctor inverter
A B Result A B Result A Result
1 1 1 1 1 1 1 0
0 0 0 1 0 1
0 1 0 0 1 1 0 1
1 0 0 0 0 0
A
A&B
IN
A
АvВ
IN
A A
Scientists and engineers have been thinking about the possibility of using logic in technology for a long time. If we examine the microcircuit under high magnification, it will amaze us with its slender architecture. From a logical point of view, electric current either flows or does not flow.
RULE FOR CONSTRUCTION OF LOGICAL DIAGRAMS:
1. Determine the number of logical variables
2. Determine the number of basic logical operations and their order
3. Draw the corresponding gate for each logical operation
4. Connect the gates in the order of logical operations.
4. Consolidation of the studied material
Task 2. Find the meaning of logical expressions:
A) F=(0v0)v(1v1) (answer 1)
B) F=(1v1)v(1v0) (answer 1)
C) F=(0&0)&(1&1) (answer 0)
Task 3: Create truth tables for the following logical expressions.
1) F=(XvY)&(XvY)
2) F=(XvY) v (X&Y)
5. Lesson summary. Evaluate the work of the class, students who excelled in the lesson.
Computer Science Lesson: Logical Operations
Goals : Introduce basic logical operations:.
Tasks :
- To form among students the concept of “logical operation”;
- Contribute to the formation logical thinking, interest in the material being studied.
Expected learning outcomes:
Students should know:
- logical operations:inversion, conjunction, disjunction, implication, equivalence;
- truth tables of logical operations;
- designation of logical operations;
- priority of logical operations.
Students should be able to:
- determine the procedure for calculating the value of a logical expression;
- construct simple and complex statements.
During the classes
I. Organizational moment.
II. Checking homework.
III. Presentation of new material.
In propositional algebra, logical operations can be performed on propositions, which result in new, composite (complex) propositions.
Def.1 Logical operation- a method of constructing a complex statement from given statements, in which the truth value of the complex statement is completely determined by the truth values of the original statements.
Let's consider three basic logical operations - inversion, conjunction, disjunction and additional ones - implication and equivalence.
Logical operation | Name | Identification by signs | Truth table | Definition |
|||||||||||||||
Inversion | Logical negation |
| The inverse of a Boolean variable is true if the variable is false, and conversely, the inverse is false if the variable is true. |
||||||||||||||||
Conjunction | Logical multiplication |
| The conjunction of two logical variables is true if and only if both statements are true |
||||||||||||||||
Disjunction | Logical addition |
| A disjunction of two logical variables is false if and only if both statements are false. |
||||||||||||||||
Implication | Logical sequence | A - condition B - consequence |
| An implication of two logical variables is false if and only if a false consequence follows from a true reason. |
|||||||||||||||
Equivalence | Logical equality |
| The equivalence of two logical variables is true if and only if both statements are simultaneously either false or true |
Exercise 1. Two simple statements are given:
A = “Pike – fish”;
B = “The crow is a songbird.”
Make up all possible compound (complex) statements from them and determine their truth.
When calculating the value of a logical expression (formula), logical operations are calculated in a certain order, according to their priority:
- inversion
- conjunction
- disjunction
- implication and equivalence
Operations of the same priority are performed from left to right. Parentheses are used to change the order of actions.
For example: given the formula.
Calculation order:
Inversion
- conjunction
- disjunction
- implication
- equivalence.
Exercise 2.
The formula is given . Determine the order of calculation.
IV. Consolidation of the studied material.
1. Among the following statements, indicate the compound ones, highlight the simple ones in them, and label each of them with a letter. Write down each compound statement using logical operations.
- The number 456 is three-digit and even.
- It is not true that the Sun moves around the Earth.
- A number is divisible by 9 if and only if the sum of its digits is divisible by 9.
- The Moon is the Earth's satellite.
- During a chemistry lesson, students performed laboratory work, and the research results were recorded in a notebook.
- If a number ends in 0, then it is divisible by 10.
- For the weather to be sunny, it is enough that there is no wind or rain.
- If I have free time and if it doesn’t rain, I won’t write essays, but I’ll go to the disco.
- If a person from childhood and youth did not allow his nerves to rule over him, then they will not get used to being irritated, and will be obedient to him.
2. Construct the negation of the following statements.
- It's dry outside.
- Today is a day off.
- Vanya was not ready for lessons today.
- It is not true that the number 3 is not a divisor of the number 198.
- Some mammals do not live on land.
- It is not true that 17 is a prime number.
3. From every three, choose a pair of statements that are negations of each other.
- “The Moon is a satellite of the Earth”, “It is not true that the Moon is a satellite of the Earth”, “It is not true that the Moon is not a satellite of the Earth”;
- “2007 2008”, “2007 ? 2008”;
- “Line a is perpendicular to line c”; “The line a is not parallel to the line c”; “The line a does not intersect the line c.”
4. Using these forms of complex statements, write down statements in Russian.
5. Find the meanings of logical expressions:
6. Two statements are given: A = “2 x 2 = 4”, B = “2 x 2 = 5”. Obviously, A=1, B=0. Which of the statements are true?
7. Given simple statements: A= (15>13), B=(4=5), C= (7
8. For what values of the number X is the logical expression not ((X>15) or (X
- lie,
- true.
9. Which of the statements A, B must be true and which must be false for there to be a false statement?
V. Lesson summary.
Summarize the material covered, evaluate the work of active students.
VI. Homework.
1. Learn definitions, know notations.
2. Statements given:
A = (The sun is shining outside),
B = (It's raining outside),
C = (The weather is cloudy outside),
D = (It's snowing outside).
Compose two complex statements, one of which will always be false in any situation, and the other true.
3. Write down a complex statement, values A, B, C take from the previous task.
Lesson 3
Teacher:Asylbekova L. S. . Grade: 8 Date: ______________
Lesson topic: Logic and logical operations.
Lesson objectives:
1. form ideas: about basic logical functions (conjunction, disjunction, implication, equivalence, negation) and truth tables of logical functions; teach students to construct truth tables of logical functions.
2. develop independence when working with logical functions when constructing truth tables.
3. attentiveness, concentration, accuracy when constructing truth tables; responsibility and self-demandingness.
During the classes
Organizing time.
Call stage.
Students are asked to complete parts of the cluster on the topic “Logical functions. Truth tables of logical functions."
The teacher updates previously acquired knowledge, which will help more effective learning of the material through questions:
Which keyword our topic?
What is the principle of cluster levels?
What is on the first, second, third level?
What level are you having problems with?
What have you heard or already know about logical elements, implementing basic logical operations?
Fill out a table on the topic of the lesson.
Conception stage.
Summarize what is the purpose of our lesson today?
The teacher summarizes the students' statements with a demonstration of presentations. Purpose of the demonstration: to form an idea of the truth table complex function, consider the algorithm for compiling a truth table, develop the ability to compile truth tables.
According to explanatory dictionary, truth table - This tabular representation of the logical circuit (operations), which lists all possible combinations of the truth values of the input signals (operands) along with the truth values of the output signal (result of the operation) for each of these combinations.
Problematic question:
Why create truth tables of logical functions?
For tabular representation of a logical diagram.
Conjunction - corresponds to union and, logical multiplication.
Disjunction - corresponds to a conjunction or logical addition.
Implication – corresponds to the conjunction if...then
Equivalence - matches the word equivalent
Negation - corresponds to the conjunction not.
Truth table.
| AIN | ||||
| AIN | ||||
4. Consolidation of practical skills.
Exercise. Determine whether the statement is true.
A) AB→AB with A-and B-l
B) ͞АВ→А῀А with A-l B-i
B) ͞͞AB→C͞D῀U with A-i B-l S-i D-l U-i
D) (A→B)῀(AB῀͞A) with A-and B-l
D) (X῀͞U) (A→B) with X-l U-i V-l A-i
5. Summing up.
Students are encouraged to carry out mutual verification solutions logical problems.
For each correct answer 1 point is awarded.
5 points – “5”
4 points – “4”
3 points – “3”
3 points – “2”
6. Reflection.
When conducting reflection, the “Sinquain” technique is used.
Sinkwine
1 I line - one noun.
2 I line - two adjectives.
3 I line - three verbs.
4 I line - one complete sentence (statement).
5 I line - one final word.
7.Assign homework.
- The concept of the science of "Logic".
- Logical operations.
- Logics.
Teacher: Deryabina I. N.
The concept of the science "Logic"
The purpose of the lesson: give the basic concepts of logic, consider the main stages of the development of logic as a science.
During the classes:
Explanation of new material:
Word logics denotes a set of rules to which the process of thinking is subject or denotes the science of the rules of reasoning and the forms in which it is carried out. Logic studies abstract thinking as a means of understanding the objective world, explores the forms and laws in which the world is reflected in the process of thinking. Main forms abstract thinking are:
- CONCEPTS
- JUDGMENTS,
- CONCLUSIONS.
CONCEPT- a form of thinking that reflects essential features a separate item or class of homogeneous items: briefcase trapeze hurricane wind
JUDGMENT- a thought in which something is affirmed or denied about objects. Propositions are declarative sentences, true or false. They can be simple or complex: Spring has come and the rooks have arrived.
CONCLUSION- a method of thinking through which new knowledge is obtained from initial knowledge; from one or more true judgments, called premises, we obtain a conclusion according to certain rules of inference. There are several types of inferences. All metals are simple substances. Lithium is a metal. Lithium is a simple substance.
To achieve truth through inference, one must follow the laws of logic.
FORMAL LOGIC- the science of the laws and forms of correct thinking.
MATHEMATICAL LOGIC studies the logical connections and relationships underlying deductive (logical) inference. (Which writer’s books talk well about the deductive method?)
Formal logic is concerned with the analysis of our ordinary meaningful inferences expressed in colloquial language. Mathematical logic studies only inferences with strictly defined objects and judgments, for which it is possible to unambiguously decide whether they are true or false.
Stages of logic development
The 1st stage is associated with the works of the scientist and philosopher Aristotle (384-322 BC). He tried to find the answer to the question “how we reason” and studied the “rules of thinking.” Aristotle was the first to give a systematic presentation of logic. He analyzed human thinking, its forms - concept, judgment, inference, and examined thinking from the side of structure, structure, that is, from the formal side. This is how formal logic arose.
Stage 2 - the emergence of mathematical or symbolic logic. Its foundations were laid by a German scientist and philosopher Gottfried Wilhelm Leibniz(1646-1716). He tried to build the first logical calculus, believed that it was possible to replace simple reasoning with actions with signs, and gave rules. But Leibniz only expressed the idea, and the Englishman finally developed it George Boole(1815-1864). Boole is considered the founder of mathematical logic as independent discipline. In his works, logic acquired its own alphabet, its own spelling and grammar. It is not for nothing that the initial section of mathematical logic is called the algebra of logic, or Boolean algebra. (you can give a home message based on the stages of logic development)
d/z notes, report on the investigation of Sherlock Holmes
Algebra of logic. Basic concepts. Scope of application of algebra-logic. Logic functions. Truth tables.
Target: Reinforce the knowledge gained in the previous lesson, give the concept of conjunction, disjunction, inversion.
During the classes:
Survey.
- Stages of development of logic.
- Basic forms of abstract thinking.
- Logic F.L, M.L.
Explanation of new material:
The basis of the operation of the logical circuit and devices is P.K. logic. In logic, a proposition is a statement - a declarative sentence - true or false.
2+8<5
5*5=25
2*2=5
A square is a parallelogram
A parallelogram is a square. -simple.
Complex (using connectives and, or and particles not.)
In M. L. the specific content of a statement is not considered, only whether it is true or false is important, therefore a statement can be represented by some ~ quantity, the value of which can be 0 or 1
0 is false, 1 is true.
For ease of recording, the statement is denoted in Latin letters. A cat has 4 legs A=1.
Moscow is located on 2 hills B=0
A PC device that performs an action on binary numbers can be considered as a kind of functional converter, with the input numbers being the values of input logical variables, and the output number being the value of a logical function, which is obtained as a result of performing certain operations. Thus, this converter implements some logical function.
The values of logical functions for different combinations of values of input variables (sets of input ~) are usually specified by a special table - a truth table.
The number of input sets ~ (Q) is determined by the expression: (Q)=2n – where n is the number of input ~ . the truth table can look like
X Y Z F (x, y, z)
0 0 0 1
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 0
d/z notes
Logical operations
The purpose of the lesson: introduce students to basic logical operations and the priority of actions in logical expressions, truth tables, learn to compile truth tables for a logical expression.
During the classes:
Survey:
The task is on the board:
In the complex statement below, highlight the simple ones. Write down a complex statement with a formula and provide a truth table:
- All planets in the solar system are spherical in shape and revolve around the sun.
- We'll go for a walk in the park or go out of town.
Questions on the spot:
- What is logic as a science?
- Formal logic and mathematics
- Examples of the deductive method
- Forms of abstract thinking
- What is a statement, what types of statements are there?
Explanation of new material:
In propositional algebra, any logical function can be expressed through basic logical operations, written as a logical expression and simplified by applying the laws of logic and properties of logical operations. Using the formula of a logical function, it is easy to calculate its truth table. It is only necessary to take into account the order of execution of logical operations (precedence) and parentheses. Operations in a logical expression are performed from left to right, taking into account parentheses. Priority of logical operations:
- INVERSION,
- CONJUNCTION,
- DISJUNCTION
CONJUNCTION
Conjunction: corresponds to the conjunction: “and”, denoted by the sign^, denotes logical multiplication.
A conjunction of two logical ~ is true if and only if both statements are true. Can be generalized for any number of variables A^B^C = 1 if A=1, B=1, C=1.
DISJUNCTION
The logical operation corresponds to the union OR, denoted by the sign v, otherwise called LOGICAL ADDITION.
A disjunction of two logical variables is false if and a pebble is false if both statements are false.
This definition can be generalized to any number of logical variables combined by a disjunction.
A v B v C = 0 only if A = O, B = O, C - 0.
The truth table of a disjunction has the following form:
INVERSION
The logical operation corresponds to the particle not, denoted ¬ or ¯ and is a logical negation.
The inverse of a boolean variable is true if the variable is false and vice versa: the inverse is false if the variable is true.
A ¬A
1 0
0 1
Statements whose truth tables coincide are called equivalent.
IMPLICATION and EQUIVALENCE
The implication “if A then B” is denoted A → B
A B A → B
0 0 1
0 1 1
1 0 0
1 1 1
The equivalence “And then B and only then” is denoted by A ~ B
A B A~ B
0 0 1
0 1 0
1 0 0
1 1 1
Fastening:
- Determine the truth table of a logical function: F (A, B, C) = A v (C ^ B), Determine the number of rows in the table: Q = 23 = 8
- Determine the number of logical operations (3) and the sequence of their execution
- We determine the number of columns: three variables + three logical operations = 6.
At the blackboard
Construct a truth table for the statements “Sasha did not complete the task” and “Sasha was reprimanded”
Sasha did not complete the task |
Sasha was reprimanded |
Result |
S/r by cards
d/z: notes
Use of propositional logic in technology. Logic circuits on contact elements.
Goal: to show the application of the topic in practice, to learn how to compose functions that describe the state of electrical circuits.
During the classes:
A logic gate is a circuit that implements the logical operations and, or, not. Let's consider the implementation of logical elements through electrical contact circuits, familiar to you from the school physics course. Contacts on the circuits will be denoted by Latin letters.
- Serial connection of contacts
- Parallel connection of contacts
Let's make a table of the dependence of the state of the circuits on all possible combinations of the state of the contacts. Let us introduce some notation. 1 - contact is closed, there is current in the circuit; 0 - contact is open, there is no current in the circuit.
Condition of a circuit with a serial connection |
Parallel Circuit Condition |
||
As you can see, a circuit with a serial connection corresponds to a logical operation and, since current in the circuit appears only when contacts A and B are closed simultaneously. A circuit with a parallel connection corresponds to a logical operation or, since current in the circuit appears as if one of the contacts is closed contacts A or B, and when they are closed simultaneously. The logical operation is not implemented through the contact circuit of an electromagnetic relay, the operating principle of which is studied in a school physics course. The non-X contact is called the inversion of the X contact, when X is closed, non-X is open, and vice versa.
Truth table for inverted contact states
Any electrical circuit can be divided into chains of series or parallel connected contacts; let’s call them elementary.
Fastening:
Break down into elementary chains
Determine the type of elementary chains, build a truth table.
S/r by cards
D/z notes
Characteristics of logical elements.
The purpose of the lesson: Get acquainted with the schematic symbols of logical elements, learn to build and read electrical circuits using formulas.
During the classes:
Explanation of new material:
ELEMENT “AND” has several inputs and 1 output, implements the logical operation “AND”
The "OR" ELEMENT has several inputs and 1 output, implements the logical operation "OR" (adder)
The “NOT” ELEMENT has 1 input and 1 output, implements the logical operation “NOT” since the output signal is always opposite to the input signal, the “NOT” element is called “inverter”
Fastening: Using cards, 1 disassemble the diagram together with the students at the board (write down a logical function according to this diagram), then independently on the spot using individual schemes.
s/r by cards
d/z: notes
Analysis, simplification and synthesis of contact circuits.
The purpose of the lesson: consolidate knowledge on the topic “Contact circuits”.
During the classes:
Repetition: On the spot, each person uses a card to break the electrical circuit into elementary chains and compile a formula for the logical function
Explanation of new material:
The main work on the electrical circuit consists of:
A) in the analysis of a contact circuit - determination of all possible conditions for the flow of electric current. This comes down to defining a logical function corresponding to this circuit
X Y not X not X v Y X ^ (not X v Y)
1 0 0 0 0
1 1 0 1 1
0 1 1 1 0
0 0 1 1 0
b) simplifying a contact circuit comes down to simplifying the corresponding formula using the laws of logic.
X ^ (not X v Y)= X ^ Y, i.e. we removed 1 contact
V) in the synthesis of a contact circuit - the development of a circuit whose operating condition is specified by a truth table or verbal description.
A B F
0 0 0
0 1 1 not A and B
or
1 0 1 A and not B
or
1 1 1 A and B
F(A,B)=(not A ^ B) v (A ^ not B) v (A ^ B)= A v B after simplification.
Fastening:
A B C F
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
F= (A ^ not B ^C) v (A ^ B ^ not C) v (A ^ B ^ C)= A ^ (B v C)
s/r by cards
d/z: notes
Logics
The purpose of the lesson: summarize knowledge on the topic “Logic”, repeat the basic parameters, prepare for the test.
During the classes:
Problem solving
A) In the statement below, highlight the simple ones. Write down complex statements in the form of a formula, provide truth tables.
Spring has come, and the rooks have arrived.
A B F
1 0 0
0 1 0
0 0 0
1 1 1
b) For the given formula, give 2 statements
not B or C
V) In accordance with the laws of logic, determine the result:
- It is not true that there is a pen on the table or a pencil is on the table
not (A or B) = not A and not B - tomorrow there will be a blizzard and it will rain or tomorrow there will be no blizzard and it will rain
(A and B) or (not A and B)=B and (not A or B)= B and 1= B - it is not true that Yura did not do this
=
A = A
G) select all elementary chains and write down the function, compile a truth table.
_ _ _ _
F(A,B,C)= A^(A V B V C) ^ B ^ C V (A V B) ^ C ^ (A V B)
A B C F
1 1 1 0
1 0 1 1
1 1 0 1
1 0 0 0
0 1 1 0
0 0 1 0
0 1 0 0
0 0 0 1
e) write down the output signal formula
F(X,Y,Z)= (X V Y V Z) ^ (Y V X) ^ (Z V Y)
D/z: draw up a truth table for the resulting formula, prepare for the test. In the statement below, highlight the simple ones. troll work.
Back forward
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Homework is checked in class using the author's test developed in the MyTest testing shell ( Annex 1), where the test is checked automatically (test results are immediately sent to the teacher’s computer).
In the study of a new topic, the definition of simple and complex statements is given, and logical operations are also considered. The explanation of new material is carried out using an interactive presentation. In order to consolidate skills and abilities, students are offered cards to fill out ( Appendix 2).
At the end of the lesson, students are asked to evaluate the degree of satisfaction with the process and the result of their work and are given cards for completing homework ( Appendix 3).
Textbook edited by Professor N.V. Makarova "Informatics and ICT".
Target:
- Study theoretical material on the topic “Logical expressions and logical operations”
- Develop logical thinking, the ability to communicate, compare and apply acquired skills in practice.
- To develop students’ cognitive activity and ability to analyze.
Lesson type: combined lesson.
Forms of work: frontal.
Visibility and equipment:
- computer;
- multimedia projector;
- presentation prepared in MS PowerPoint;
- test on the topic “Basic concepts of algebra of logic” ;
- cards for consolidating the material covered;
- homework card.
Lesson plan:
- Organizing time (1 min.)
- Checking the material studied (10 min.)
- Learning new material (20 minutes.)
- Consolidation of the studied material (oral work, 5 minutes.)
- Summing up the lesson (2 minutes.)
- Homework (2 minutes.)
During the classes
1. Organizational moment.
Purpose: to prepare students for the lesson.
The topic of the lesson is announced. The students are given a task: to show how they have learned to solve problems on the topic.
2. Repetition of the studied material.
Executing a test on the topic “Basic concepts of logical algebra” in the MyTest testing shell. (Appendix 1.mtf)
3. Studying new material.
Questions to study:
- Simple and complex expressions.
- Basic logical operations.
When explaining new material, a computer presentation is used (presentation.PPT)
- 1. Simple and complex expressions.
Logical expressions can be simple or complex.
A simple logical expression consists of one statement and does not contain logical operations. In a simple Boolean expression, there are only two possible results - either true or false.
A complex logical expression contains statements combined with logical operations. By analogy with the concept of a function in algebra, a complex logical expression contains arguments, which are statements.
- 2. Basic logical operations.
As they explain new material, students fill out the following table in their notebooks.
| Logical operation name | Logical operation notation | Result of logical operation | Truth table | Examples |
| Negation | ||||
| Disjunction | ||||
| Conjunction | ||||
| Implication | ||||
| Equivalence |
The following are used as basic logical operations in complex logical expressions:
- NOT(logical negation, inversion);
- OR(logical addition, disjunction);
- AND(logical multiplication, conjunction)
Operation NOT - logical negation (inversion)
A logical operation is NOT applied to a single argument, which can be a simple or complex logical expression. The result of the operation is NOT the following:
- if the original expression is true, then the result of its negation will be false;
- if the original expression is false, then the result of its negation will be true.
For the negation operation NOT, the following conventions are accepted: NOT, ‾, ˥ not A. The result of the negation operation is NOT determined by the following truth table.
OR operation - logical addition (disjunction, union)
The logical OR operation performs the function of combining two statements, which can be either a simple or a complex logical expression. Statements that are the starting points for a logical operation are called arguments.
The result of the OR operation is an expression that will be true if and only if at least one of the original expressions is true.
The result of the OR operation is determined by the following truth table:
| A | IN | A v B |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Designations used: A or B; A v B; A og B. When performing complex logical transformations, for clarity, we agree to use the notation A + B, where A, B are arguments (initial statements).
Operation AND - logical multiplication (conjunction)
The logical operation AND performs the function of intersection of two statements (arguments), which can be either a simple or a complex logical expression.
The result of the AND operation is an expression that will be true if and only if both original expressions are true.
The result of the AND operation is determined by the following truth table:
| A | IN | A^B |
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Designations used: A and B; A^B; A&B; A and B.
Let us agree to use the notation A-B when performing complex logical transformations, where A, B are arguments (initial statements).
Operation "IF"- TO» - logical consequence (implication)
This operation connects two simple logical expressions, of which the first is a condition, and the second is a consequence of this condition.
Designations used:
if A, then B; A entails B; if A then B; A-»B.
The result of the operation of implication is false only if premise A is true and conclusion B (consequence) is false.
Truth table:
Operation “A if and only if B” (equivalence, equivalence)
Designation used: A ~ IN.
The result of the equivalence operation is true only if A and B are both true or false at the same time.
Truth table:
| A | IN | A ~ IN |
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
4. Consolidation of the studied material
This material is distributed to each student. (Appendix 2)
5. Summing up the lesson
Tell me, was today's lesson educational for you?
What do you remember most from the lesson?
6. Homework
- Textbook. Section 23.2., fill out the “Logical Operations” table to the end.
- Perform a task(Appendix 3)
- Prepare for testing.
- Know the answers to the questions:
- what kind of statements there are;
- which statements are called simple and which are complex;
- basic logical operations and their properties.
