Lesson outline on the topic:

"Foundations of Logic".

The purpose of the lesson: to form students' concept of forms of thinking, to form concepts: logical statement, logical values, logical operations.

Tasks:

Ø  Educational: to acquaint children with the forms of thinking, to form concepts: logical statement, logical values, logical operations.

Ø  Developing: create conditions for the development of the cognitive interest of students, contribute to the development of memory, attention, logical thinking;

Ø  Educational: contribute to the education of the ability to listen to the opinions of others, to work in a team and groups.

Lesson type: A lesson in learning and primary consolidation of new knowledge

Short description:

1. Organizational moment - 2 min.

2. Reporting the topic and setting the goals of the lesson - 2 min.

3. Learning new material - 22 min.

4. Consolidation of the acquired knowledge - 10 min.

5. Summing up the lesson - 2 min.

6. Homework - 2 min.

During the classes

I. Organizational moment

Greet the students, set them up for further work.

II. Communication of the topic and objectives of the lesson.

How does a person think?

What is a statement in our speech and what is not?

What are the similarities and differences in arithmetic multiplication and logical multiplication?

We will try to answer these and some other questions today in the lesson. We will also get acquainted with the basic logical expressions and operations, we will learn some of the components of our thinking.

So, the topic of our lesson is the Basics of Logic.

III. Explanation of the new material...

1. At the heart of modern logic are doctrines created by ancient Greek thinkers, although the first teachings about the forms and methods of thinking originated in ancient China and India. The founder of formal logic is Aristotle, who was the first to separate the logical forms of thinking from its content.

Logics is the science of forms and ways of thinking. This is the doctrine of methods of reasoning and evidence.

The laws of the world, the essence of objects, what is common in them, we learn through abstract thinking. Logic allows you to build formal models of the surrounding world, distracting from the content side.

Thinking is always carried out through concepts, statements and inferences.

Concept - This is a form of thinking that highlights the essential features of an object or class of objects, allowing them to be distinguished from others.

Example

Rectangle, pouring rain, computer.

Utterance is a formulation of your understanding of the world around you. Utterance is a declarative sentence in which something is affirmed or denied.

With regard to a statement, you can say whether it is true or false. A true statement will be in which the connection of concepts correctly reflects the properties and relations of real things. A false statement will be when it contradicts reality.

Example

True saying: "The letter 'e' is a vowel."

False saying: "The computer was invented in the middle of the 19th century."

Example

Which of the sentences are statements? Determine their truth.

1. How long is this tape?

2. Listen to the message.

3. Do your morning exercises!

4. Name the input device.

5. Who is missing?

6. Paris is the capital of England. (FALSE)

7. The number 11 is prime. (TRUE)

8.4 + 5 = 10. (FALSE)

9. You can't get a fish out of a pond without difficulty.

10. Add the numbers 2 and 5.

11. Some bears live in the north. (TRUE)

12. All bears are brown. (FALSE)

13. What is the distance from Moscow to Leningrad.

Inference - This is a form of thinking, with the help of which a new judgment (knowledge or conclusion) can be obtained from one or several judgments.

Example

The saying is given: "All angles of an isosceles triangle are equal." Get the saying "This triangle is equilateral" by reasoning. Let the base of the triangle be side c, then a = b. Since all angles in a triangle are equal, therefore, the base can be any other side, for example a. Then b = c. Therefore, a = b = c. The triangle is equilateral.

2. Logical expressions and operations

Algebra is the science of general operations similar to addition and multiplication, which are performed not only on numbers, but also on other mathematical objects, including statements. So! algebra is called the algebra of logic. Algebra of logic is abstracted from the semantic content of statements and takes into account only the truth or falsity of the statement.

You can define the concepts of boolean variable, boolean function, and boolean operation.

Boolean variable is a simple statement containing only one thought. Its symbolic designation is a Latin letter. The value of a logical variable can only be the constants TRUE and FALSE (1 and 0).

A compound statement is a logical function that contains several simple thoughts connected to each other using logical operations. Its symbolic designation is F (A, B, ...).

Composite statements can be built on the basis of simple statements.

Logical operations - logical action.

There are three basic logical operations - conjunction, disjunction and negation, and additional ones - implication and equivalence. If a compound statement (logical function) is expressed as a formula that includes logical variables and signs of logical operations, then you get a logical expression whose value can be calculated. The Boolean expression value can only be FALSE or TRUE. When composing a logical expression, it is necessary to take into account the order of performing logical operations, namely:

1) actions in brackets;

2) inversion, conjunction, disjunction, implication, equivalence.

Example

Write down the following statement in the form of a logical expression: "In the summer, Petya will go to the village and, if the weather is good, he will go fishing."

1. Let's break the compound statement into simple statements: "Petya will go to the village", "The weather will be fine", "He will go fishing."

Let's denote them through boolean variables:

A = Petya will go to the village;

B = The weather will be nice;

S = He will go fishing.

2. Let's write the statement in the form of a logical expression, taking into account the order of actions. If necessary, place the brackets:

F = A & (B + C).

Let's fill in the presented table.

IV. Consolidation of the studied material.

Division of the class into groups.

Exercise 1.

There are two simple sayings:

A - "Number 10 - even";

B - "The wolf is a herbivore".

Make up all possible compound statements from them and determine their truth. Answer:

Exercise 2.

Record the following statements as logical expressions.

1. The number 17 is odd and two-digit.

2. It is not true that the cow is a predatory animal.

3. In a physics lesson, students performed laboratory work and reported the research results to the teacher.

4. If a number is divisible by 2, then it is even. Cross the street only on green light.

6. In a computer science lesson, special rules of conduct must be observed.

7. When water freezes, heat is generated.

8. If Masha is Sasha's sister, then Sasha is Masha's brother.

9. If the computer is turned on, you can work on it.

10. A driver's license can be obtained if and only when you turn 18.

11. The computer performs calculations if it is turned on.

12. You can buy groceries at the store if you have money.

13. The quieter you go - the further you will be.

Exercise 3.

Make and write down true complex statements from simple ones using logical operations.

1. It is not true that 10> Y> 5 and Z <0 (answer: (Y <10) & (Y> 5) & (Z <0).

2. Z is min (Z, Y) (answer: Z <Y).

3. A is max (A, B, C) (answer: (A> B) & (A> C)).

4. Any of the numbers X, Y, Z is positive (answer: (X> 0) v (Y> 0) v (Z> 0).

5. Any of the numbers X, Y, Z is negative (answer: (X <0) v (Y <0) v (Z <0).

6. At least one of the numbers K, L, M is not negative (answer: (K> 0) v (I> 0) v (M> O))

7. At least one of the numbers X, Y, Z is not less than 12 (answer: (X> 12) v (Y> 12) v (Z> 12))

8. All numbers X, Y, Z are 12 (answer: (X = 12) & (Y = 12) & (Z = 12)).

Exercise 4.

Find the values ​​of the boolean expressions:

F = (0v0) v (lvl) (answer: 1)

F = (lvl) v (lv0) (answer: 1)

F = (0 & 0) & (1 & 1) (answer: 0)

F = ¬1 & (1 v1) v (¬0 & 1) (answer: 1)

F = (¬1v1) & (1v¬1) & (¬1v0) (answer: 0)

Vi. Summing up the lesson.

Frontal conversation with students on the topic of the lesson.

Grading.

Vii. Homework

Knowledge level: learn basic definitions, know notation.

Understanding level:

Problem 1

From two simple statements, construct a complex statement using logical connectives "AND", "OR". Write down logical statements using logical operations and determine their truth.

1. Andrey is older than Sveta. Natasha is older than Sveta.

2. One tenth grade goes on an excursion to the museum. The second tenth grade goes to the theater.

3. There are textbooks on the shelf. There are reference books on the shelf.

4. Some of the children are girls. The rest are boys.

Problem 2

For logical expressions, formulate compound statements in ordinary language:

1) (Y> 1 and Y <3) or (Y <8n Y> 4)

2) (X = Y) and (X = Z)

3) Not (X <0) and X <10 or (Y> 0)

4) (0 <X) and (X <5) and (not (Y <10))

Problem 3

What logical expression corresponds to the statement: "Point X belongs to the interval (A; B)".

1) (X <A) or (X> B)

2) (X> A) and (X <B)

3) Not (X <A) or (X <B)

4) (X> A) or (X> B)

Application Level: Give examples of compound statements from the following school subjects and write them down using logical operations:

1) biology; 5) literature;

2) geography; 6) mathematics;

3) informatics; 7) the Russian language.

4) history;