Solutions. Logic and true sets

Catalog of tasks.
Number of programs with a mandatory stage

Performer A16 converts the number written on the screen.

The performer has three teams, which are assigned numbers:

1. Add 1

2. Add 2

3. Multiply by 2

The first of them increases the number on the screen by 1, the second increases it by 2, the third multiplies it by 2.

The program for performer A16 is a sequence of commands.

How many programs are there that convert the original number 3 into the number 12 and at the same time the program's calculation path contains the number 10?

A program's computation trajectory is a sequence of results from the execution of all program commands. For example, for program 132 with the initial number 7, the trajectory will consist of the numbers 8, 16, 18.

Solution.

The required number of programs is equal to the product of the number of programs that obtain the number 10 from the number 3 by the number of programs that obtain the number 12 from the number 10.

Let R(n) be the number of programs that convert the number 3 into the number n, and P(n) be the number of programs that convert the number 10 into the number n.

For all n > 5 the following relations are true:

1. If n is not divisible by 2, then R(n) = R(n - 1) + R(n - 2), since there are two ways to obtain n - by adding one or adding two. Similarly P(n) = P(n - 1) + P(n - 2)

2. If n is divisible by 2, then R(n) = R(n - 1) + R(n - 2) + R(n / 2). Similarly P(n) = P(n - 1) + P(n - 2) + P(n / 2)

Let us sequentially calculate the values ​​of R(n):

R(5) = R(4) + R(3) = 1 + 1 = 2

R(6) = R(5) + R(4) + R(3) = 2 + 1 + 1 = 4

R(7) = R(6) + R(5) = 4 + 2 = 6

R(8) = R(7) + R(6) + R(4) = 6 + 4 + 1 = 11

R(9) = R(8) + R(7) = 11 + 6 = 17

R(10) = R(9) + R(8) + R(5) = 17 + 11 + 2 = 30

Now let's calculate the values ​​of P(n):

P(11) = P(10) = 1

P(12) = P(11) + P(10) = 2

Thus, the number of programs that satisfy the conditions of the problem is 30 · 2 = 60.

Answer: 60.

Answer: 60

Source: Demo version of the Unified State Exam 2017 in computer science.

1. Add 1

2. Add 3

How many programs are there for which, given the initial number 1, the result is the number 17 and at the same time the computation path contains the number 9? A program's computation trajectory is a sequence of results from the execution of all program commands. For example, for program 121 with the initial number 7, the trajectory will consist of the numbers 8, 11, 12.

Solution.

We use the dynamic programming method. let's create an array dp, where dp[i] is the number of ways to get the number i using such commands.

Dynamics base:

Transition formula:

dp[i]=dp + dp

This does not take into account the values ​​for numbers greater than 9, which can be obtained from numbers less than 9 (thus skipping the trajectory of 9):

Answer: 169.

Answer: 169

Source: Training work in COMPUTER SCIENCE, grade 11 November 29, 2016 Option IN10203

Performer May17 converts the number on the screen.

The performer has two teams, which are assigned numbers:

1. Add 1

2. Add 3

The first command increases the number on the screen by 1, the second increases it by 3. The program for the May17 performer is a sequence of commands.

How many programs are there for which, given the initial number 1, the result is the number 15 and at the same time the computation trajectory contains the number 8? A program's computation trajectory is a sequence of results from the execution of all program commands. For example, for program 121 with the initial number 7, the trajectory will consist of the numbers 8, 11, 12.

Solution.

We use the dynamic programming method. Let's create an array dp, where dp[i] is the number of ways to get the number i using such commands.

Dynamics base:

Transition formula:

dp[i]=dp + dp

But this does not take into account numbers that are greater than 8, but we can get to them from a value less than 8. The following will show the values ​​​​in cells dp from 1 to 15: 1 1 1 2 3 4 6 9 9 9 18 27 36 54 81 .

Let's first define what we have in the problem:

• a logical function F defined by some expression. The elements of the truth table of this function are also presented in the problem in the form of a table. Thus, when substituting specific values ​​of x, y, z from the table into the expression, the result should coincide with the one given in the table (see explanation below).
• The variables x, y, z and the three columns that correspond to them. Moreover, in this problem we do not know which column corresponds to which variable. That is, in the column Variable. 1 can be either x, y or z.
• We are asked to determine which column corresponds to which variable.

Let's look at an example.

Solution

1. Let's return now to the solution. Let's take a closer look at the formula: \((\neg z) \wedge x \vee x\wedge y\)
2. It contains two constructions with a conjunction, connected by a disjunction. As is known, most often the disjunction is true (for this it is enough that one of the terms is true).
3. Let's then look carefully at the lines where the expression F is false.
4. The first line is not interesting to us, since it does not determine where what is (all values ​​are the same).
5. Let us then consider the penultimate line, it contains most of 1, but the result is 0.
6. Can z be in the third column? No, because in this case there will be 1s everywhere in the formula, and, therefore, the result will be equal to 1, but according to the truth table, the value of F in this row is 0. Therefore, z cannot be Variable. 3.
7. Similarly, for the previous line we have that z cannot be Variable. 2.
8. Hence, z is Variable. 1.
9. Knowing that z is in the first column, consider the third row. Can x be in the second column? Let's substitute the values:
   \((\neg z) \wedge x \vee x\wedge y = \\ = (\neg 0) \wedge 1 \vee 1\wedge 0 = \\ = 1 \wedge 1 \vee 0 = \\ = 1 \vee 0 = 1\)
10. However, according to the truth table, the result must be 0.
11. Hence, x cannot be Per. 2.
12. Hence, x is Variable. 3.
13. Therefore, by the method of elimination, y is Variable. 2.
14. Thus, the answer is as follows: zyx (z - Variable 1, y - Variable 2, x - Variable 3).​

Demonstration version of the Unified State Exam 2019 – task No. 2

Misha filled out the truth table of the function (¬x /\ ¬y) \/ (y≡z) \/ ¬w, but only managed to fill out a fragment of three different lines, without even indicating which column of the table corresponds to each of the variables w, x ,
y, z.

Determine which table column each variable w, x, y, z corresponds to.
In your answer, write the letters w, x, y, z in the order in which their corresponding columns appear (first the letter corresponding to the first column; then the letter corresponding to the second column, etc.). Letters
In your answer, write in a row; there is no need to put any separators between the letters.
Example. If the function were given by the expression ¬x \/ y, depending on two variables, and the table fragment would look like

then the first column would correspond to the variable y, and the second column would correspond to the variable x. The answer should have been written yx.

(¬x ¬y)+(y≡z)+¬w=0

w=1 w must be true; w - last

y and z must be different, so before the last one, it's x. the first two are y and z or z and y.

y and x cannot be false at the same time. The first is z.

Answer: zyxw

Demonstration version of the Unified State Exam 2018 - task No. 2

The logical function F is given by the expression ¬x \/ y \/ (¬z /\ w). The figure shows a fragment of the truth table of the function F, containing all sets of arguments for which the function F is false. Determine which column of the truth table of function F corresponds to each of the variables w, x, y, z

In your answer, write the letters w, x, y, z in the order in which their corresponding columns appear (first - the letter corresponding to the first column; then - the letter corresponding to the second column, etc.) Write the letters in the answer in a row, There is no need to put any separators between letters. Example. If the function were given by the expression ¬x\/y, depending on two variables: x and y, and a fragment of its truth table was given, containing all sets of arguments for which the function is true.

Then the first column would correspond to the variable y, and the second column would correspond to the variable x. The answer should have been written: yx.

Answer: xzwy

Logic function F is given by the expression x/\ ¬y/\ (¬z\/ w).

The figure shows a fragment of the truth table of the function F containing All sets of arguments for which the function F true.

Determine which column of the function's truth table F each of the variables corresponds w, x, y, z.

Write the letters in your answer w, x, y, z in the order they come

their corresponding columns (first – the letter corresponding to the first

column; then the letter corresponding to the second column, etc.) Letters

In your answer, write in a row, put no separators between letters.

no need.

Demonstration version of the Unified State Exam 2017 - task No. 2

Solution:

A conjunction (logical multiplication) is true if and only if all statements are true. Therefore the variable X 1 .

Variable ¬y must match the column in which all values ​​are equal 0 .

A disjunction (logical addition) of two statements is true if and only if at least one statement is true.
Disjunction ¬z\/y z=0, w=1.

Thus, the variable ¬z w corresponds to the column with variable 4 (column 4).

Answer: zyxw

Demonstration version of the Unified State Exam 2016 - task No. 2

Logic function F is given by the expression (¬z)/\x \/ x/\y. Determine which column of the truth table of the function F corresponds to each of the variables x, y, z.

In your answer, write the letters x, y, z in the order in which their corresponding columns appear (first - the letter corresponding to the 1st column; then - the letter corresponding to the 2nd column; then - the letter corresponding to the 3rd column) . Write the letters in the answer in a row; there is no need to put any separators between the letters.

Example. Let an expression x → y be given, depending on two variables x and y, and a truth table:

Then the 1st column corresponds to the variable y, and the 2nd column
corresponds to the variable x. In the answer you need to write: yx.

Solution:

1. Let's write it down for this expression in simpler notation:

¬z*x + x*y = x*(¬z + y)

2. Conjunction (logical multiplication) is true if and only if all statements are true. Therefore, so that the function ( F) was equal to one ( 1 ), each factor must be equal to one ( 1 ). Thus, when F=1, variable X must match the column in which all values ​​are equal 1 .

3. Consider (¬z + y), at F=1 this expression is also equal to 1 (see point 2).

4. Disjunction (logical addition) of two statements is true if and only if at least one statement is true.
Disjunction ¬z\/y in this line will be true only if

1. z = 0; y = 0 or y = 1;
2. z = 1; y = 1

5. Thus, the variable ¬z corresponds to column with variable 1 (1 column), variable y

Answer: zyx

KIM Unified State Exam 2016 (early period)– task No. 2

The logical function F is given by the expression

(x /\ y /\¬z) \/ (x /\ y /\ z) \/ (x /\¬y /\¬z).

The figure shows a fragment of the truth table of the function F, containing all sets of arguments for which the function F is true. Determine which column of the truth table of the function F corresponds to each of the variables x, y, z.

In your answer, write the letters x, y, z in the order in which their corresponding columns appear (first - the letter corresponding to the first column; then - the letter corresponding to the second column, etc.) Write the letters in the answer in a row, no separators There is no need to put it between letters.

R solution:

Let's write the given expression in simpler notation:

(x*y*¬z) + (x*y*z) + (x*¬y*¬z)=1

This expression is true when at least one of (x*y*¬z), (x*y*z), (x*¬y*¬z) equals 1. Conjunction (logical multiplication) is true if and only if when all statements are true.

At least one of these disjunctions x*y*¬z; x*y*z; x*¬y*¬z will be true only if x=1.

Thus, the variable X corresponds to the column with variable 2 (column 2).

Let y- variable 1, z- prem.3. Then, in the first case x*¬y*¬z will be true in the second case x*y*¬z, and in the third x*y*z.

Answer: yxz

The symbol F denotes one of the following logical expressions from three arguments: X, Y, Z. A fragment of the truth table of the expression F is given (see the table on the right). Which expression matches F?

1) X ∧ Y ∧ Z 2) ¬X ∨ Y ∨¬Z 3) X ∧ Y ∨ Z 4) X ∨ Y ∧ ¬Z

Solution:

1) X ∧ Y ∧ Z = 1.0.1 = 0 (does not match on 2nd line)

2) ¬X ∨ Y ∨¬Z = ¬0 ∨ 0 ∨ ¬0 = 1+0+1 = 1 (does not match on the 1st line)

3) X ∧ Y ∨ Z = 0.1+0 = 0 (does not match on the 3rd line)

4) X ∨ Y ∧ ¬Z (corresponds to F)

X ∨ Y ∧ ¬Z = 0 ∨ 0 ∧ ¬0 = 0+0.1 = 0

X ∨ Y ∧ ¬Z = 1 ∨ 0 ∧ ¬1 = 1+0.0 = 1

X ∨ Y ∧ ¬Z = 0 ∨ 1 ∧ ¬0 = 0+1.1 = 1

Answer: 4

Given a fragment of the truth table of the expression F. Which expression corresponds to F?

1) (A → ¬B) ∨ C 2) (¬A ∨ B) ∧ C 3) (A ∧ B) → C 4) (A ∨ B) → C

Solution:

1) (A → ¬B) ∨ C = (1 → ¬0) ∨ 0 = (1 → 1) + 0 = 1 + 0 = 1 (does not match on 2nd line)

2) (¬A ∨ B) ∧ C = (¬1 ∨ 0) ∧ 1 = (0+0).1 = 0 (does not match on the 3rd line)

3) (A ∧ B) → C = (1 ∧ 0) → 0 = 0 → 0 = 1 (does not match on 2nd line)

4) (A ∨ B) → C (corresponds to F)

(A ∨ B) → C = (0 ∨ 1) → 1 = 1

(A ∨ B) → C = (1 ∨ 0) → 0 = 0

(A ∨ B) → C = (1 ∨ 0) → 1 = 1

Answer: 4

A logical expression is given that depends on 6 logical variables:

X1 ∨ ¬X2 ∨ X3 ∨ ¬X4 ∨ X5 ∨ X6

How many different sets of variable values ​​are there for which the expression is true?

1) 1 2) 2 3) 63 4) 64

Solution:

False expression only in 1 case: X1=0, X2=1, X3=0, X4=1, X5=0, X6=0

X1 ∨ ¬X2 ∨ X3 ∨ ¬X4 ∨ X5 ∨ X6 = 0 ∨ ¬1 ∨ 0 ∨ ¬1 ∨ 0 ∨ 0 = 0

There are 2 6 =64 options in total, which means true

Answer: 63

A fragment of the truth table of the expression F is given.

Which expression matches F?

1) x1 ∨ x2 ∨ ¬x3 ∨ x4 ∨ ¬x5 ∨ x6 ∨ ¬x7
2) x1 ∨ ¬x2 ∨ x3 ∨ ¬x4 ∨ ¬x5 ∨ x6 ∨ x7
3) x1 ∧ ¬x2 ∧ x3 ∧ ¬x4 ∧ x5 ∧ ¬x6 ∧ x7
4) x1 ∧ x2 ∧ ¬x3 ∧ x4 ∧ ¬x5 ∧ x6 ∧ ¬x7

Solution:

1) x1 ∨ x2 ∨ ¬x3 ∨ x4 ∨ ¬x5 ∨ x6 ∨ ¬x7 = 0 + 1 + … = 1 (does not match on the 1st line)

2) x1 ∨ ¬x2 ∨ x3 ∨ ¬x4 ∨ ¬x5 ∨ x6 ∨ x7 = 0 + 0 + 0 + 0 + 0 + 1 + 0 = 1 (does not match on the 1st line)

3) x1 ∧ ¬x2 ∧ x3 ∧ ¬x4 ∧ x5 ∧ ¬x6 ∧ x7 = 1.0. ...= 0 (does not match on 2nd line)

4) x1 ∧ x2 ∧ ¬x3 ∧ x4 ∧ ¬x5 ∧ x6 ∧ ¬x7 (corresponds to F)

x1 ∧ x2 ∧ ¬x3 ∧ x4 ∧ ¬x5 ∧ x6 ∧ ¬x7 = 1.1.1.1.1.1.1 = 1

x1 ∧ x2 ∧ ¬x3 ∧ x4 ∧ ¬x5 ∧ x6 ∧ ¬x7 = 0. … = 0

Answer: 4

What expression can F be?

1) x1 ∧ ¬x2 ∧ x3 ∧ ¬x4 ∧ x5 ∧ x6 ∧ ¬x7 ∧ ¬x8
2) ¬x1 ∨ x2 ∨ x3 ∨ ¬x4 ∨ ¬x5 ∨ ¬x6 ∨ ¬x7 ∨ x8
3) ¬x1 ∧ x2 ∧ ¬x3 ∧ x4 ∧ x5 ∧ ¬x6 ∧ ¬x7 ∧ ¬x8
4) ¬x1 ∨ ¬x2 ∨ ¬x3 ∨ ¬x4 ∨ ¬x5 ∨ ¬x6 ∨ ¬x7 ∨ ¬x8

Solution:

1) x1 ∧ ¬x2 ∧ x3 ∧ ¬x4 ∧ x5 ∧ x6 ∧ ¬x7 ∧ ¬x8 = x1 . ¬x2. 0 . ... = 0 (does not match on 1st line)

2) ¬x1 ∨ x2 ∨ x3 ∨ ¬x4 ∨ ¬x5 ∨ ¬x6 ∨ ¬x7 ∨ x8 (corresponds to F)

3) ¬x1 ∧ x2 ∧ ¬x3 ∧ x4 ∧ x5 ∧ ¬x6 ∧ ¬x7 ∧ ¬x8 = … ¬x7 ∧ ¬x8 = … ¬1 ∧ ¬x8 = … 0 ∧ ¬x8 = 0 (does not match on 1- th line)

4) ¬x1 ∨ ¬x2 ∨ ¬x3 ∨ ¬x4 ∨ ¬x5 ∨ ¬x6 ∨ ¬x7 ∨ ¬x8 = ¬x1 ∨ ¬x2 ∨ ¬x3 … = ¬1 ∨ ¬x2 ∨ ¬0 .. = 1 (not matches on the 2nd line)

Answer: 2

Given is a fragment of the truth table for the expression F:

Find the minimum possible number of different rows in the complete truth table of this expression in which the value x5 matches F.

Solution:

Minimum possible number of distinct rows in which x5 matches F = 4

Answer: 4

Given is a fragment of the truth table for the expression F:

Find the maximum possible number of distinct rows in the complete truth table of this expression in which the value x6 does not coincide with F.

Solution:

Maximum possible number = 2 8 = 256

The maximum possible number of different rows in which the value x6 does not match F = 256 - 5 = 251

Answer: 251

Given is a fragment of the truth table for the expression F:

Find the maximum possible number of different rows of the complete truth table of this expression in which the value ¬x5 ∨ x1 coincides with F.

Solution:

1+0=1 - does not match F

0+0=0 - does not match F

0+0=0 - does not match F

0+1=1 - same as F

1+0=1 - same as F

2 7 = 128 — 3 = 125

Answer: 125

Each Boolean expression A and B depends on the same set of 6 variables. In the truth tables, each of these expressions has exactly 4 units in the value column. What is the minimum possible number of ones in the value column of the truth table of the expression A ∨ B?

Solution:

Answer: 4

Each Boolean expression A and B depends on the same set of 7 variables. In the truth tables, each of these expressions has exactly 4 units in the value column. What is the maximum possible number of ones in the value column of the truth table of the expression A ∨ B?

Solution:

Answer: 8

Each Boolean expression A and B depends on the same set of 8 variables. In the truth tables, each of these expressions has exactly 5 units in the value column. What is the minimum possible number of zeros in the value column of the truth table of the expression A ∧ B?

Solution:

2 8 = 256 — 5 = 251

Answer: 251

Each Boolean expression A and B depends on the same set of 8 variables. In the truth tables, each of these expressions has exactly 6 units in the value column. What is the maximum possible number of zeros in the value column of the truth table of the expression A ∧ B?

Solution:

Answer: 256

The Boolean expressions A and B each depend on the same set of 5 variables. There are no matching rows in the truth tables of both expressions. How many ones will be contained in the value column of the truth table of the expression A ∧ B?

Solution:

There are no matching rows in the truth tables of both expressions.

Answer: 0

The Boolean expressions A and B each depend on the same set of 6 variables. There are no matching rows in the truth tables of both expressions. How many ones will be contained in the value column of the truth table of the expression A ∨ B?

Solution:

(a . ¬c) + (¬b . ¬c)

When c is 1, F is zero so the last column is c.

To determine the first and second columns, we can use the values ​​from the 3rd row.

(a . 1) + (¬b . 1) = 0

Answer: ABC

The logical function F is given by the expression (a ∧ c)∨ (¬a ∧ (b ∨ ¬c)). Determine which column of the truth table of the function F corresponds to each of the variables a, b, c.

Based on the fact that when a=0 and c=0, then F=0, and the data from the second row, we can conclude that the third column contains b.

Answer: cab

The logical function F is given by x ∧ (¬y ∧ z ∧ ¬w ∨ y ∧ ¬z). The figure shows a fragment of the truth table of the function F, containing all sets of arguments for which the function F is true. Determine which column of the truth table of the function F corresponds to each of the variables x, y, z, w.

In your answer, write the letters x, y, z, w in the order in which their corresponding columns appear.

Solution:

x ∧ (¬y ∧ z ∧ ¬w ∨ y ∧ ¬z)

x. (¬y . z . ¬w . y . ¬z)

Based on the fact that at x=0, then F=0, we can conclude that the second column contains x.

Answer: wxzy

Job source: Solution 2437. Unified State Exam 2017. Computer Science. V.R. Leschiner. 10 options.

Task 2. The logical function F is given by the expression . Determine which column of the truth table of the function F corresponds to each of the variables x, y, z.

In your answer, write the letters x, y, z in the order in which their corresponding columns appear (first - the letter corresponding to the 1st column, then - the letter corresponding to the 2nd column, then - the letter corresponding to the 3rd column) . Write the letters in the answer in a row; there is no need to put any separators between the letters.

Solution.

Let us rewrite the expression for F taking into account the priorities of the operations of negation, conjunction and disjunction:

.

Consider the 4th row of the table (1,1,0)=0. From this we can see that the third place must be either the variable y or the variable z, otherwise the second bracket will contain 1, which will lead to the value F=1. Now consider the 5th row of the table (0,0,1)=1. Since x must be in the first or second place, the first parenthesis will give 1 only when y is in the 3rd place. Considering that the second bracket is always equal to 0, then F=1 is obtained due to the 1 in the first bracket. Thus, we found that y is in 3rd place. Finally, consider the 7th row of the table (1,0,1)=0. Here y=1 and for F=0 it is necessary to have z=0 and x=1, therefore, x is in the 1st place, and z is in the second.

Logic function F is given by the expression x/\ ¬y/\ (¬z\/ w).

The figure shows a fragment of the truth table of the function F containing All sets of arguments for which the function F true.

Determine which column of the function's truth table F each of the variables corresponds w, x, y, z.

Write the letters in your answer w, x, y, z in the order they come

their corresponding columns (first – the letter corresponding to the first

column; then the letter corresponding to the second column, etc.) Letters

In your answer, write in a row, put no separators between letters.

no need.

Demo version of the Unified State Examination USE 2017 – task No. 2

Solution:

A conjunction (logical multiplication) is true if and only if all statements are true. Therefore the variable X 1 .

Variable ¬y must match the column in which all values ​​are equal 0 .

A disjunction (logical addition) of two statements is true if and only if at least one statement is true.
Disjunction ¬z\/y z=0, w=1.

Thus, the variable ¬z w corresponds to the column with variable 4 (column 4).

Answer: zyxw

Demo version of the Unified State Examination USE 2016 – task No. 2

Logic function F is given by the expression (¬z)/\x \/ x/\y. Determine which column of the truth table of the function F corresponds to each of the variables x, y, z.

In your answer, write the letters x, y, z in the order in which their corresponding columns appear (first - the letter corresponding to the 1st column; then - the letter corresponding to the 2nd column; then - the letter corresponding to the 3rd column) . Write the letters in the answer in a row; there is no need to put any separators between the letters.

Example. Let an expression x → y be given, depending on two variables x and y, and a truth table:

Then the 1st column corresponds to the variable y, and the 2nd column
corresponds to the variable x. In the answer you need to write: yx.

Solution:

1. Let's write the given expression in simpler notation:

¬z*x + x*y = x*(¬z + y)

2. Conjunction (logical multiplication) is true if and only if all statements are true. Therefore, so that the function ( F) was equal to one ( 1 ), each factor must be equal to one ( 1 ). Thus, when F=1, variable X must match the column in which all values ​​are equal 1 .

3. Consider (¬z + y), at F=1 this expression is also equal to 1 (see point 2).

4. Disjunction (logical addition) of two statements is true if and only if at least one statement is true.
Disjunction ¬z\/y in this line will be true only if

1. z = 0; y = 0 or y = 1;
2. z = 1; y = 1

5. Thus, the variable ¬z corresponds to column with variable 1 (1 column), variable y

Answer: zyx

KIM Unified State Examination Unified State Exam 2016 (early period)– task No. 2

The logical function F is given by the expression

(x /\ y /\¬z) \/ (x /\ y /\ z) \/ (x /\¬y /\¬z).

The figure shows a fragment of the truth table of the function F, containing all sets of arguments for which the function F is true. Determine which column of the truth table of the function F corresponds to each of the variables x, y, z.

In your answer, write the letters x, y, z in the order in which their corresponding columns appear (first - the letter corresponding to the first column; then - the letter corresponding to the second column, etc.) Write the letters in the answer in a row, no separators There is no need to put it between letters.

R solution:

Let's write the given expression in simpler notation:

(x*y*¬z) + (x*y*z) + (x*¬y*¬z)=1

This expression is true when at least one of (x*y*¬z), (x*y*z), (x*¬y*¬z) equals 1. Conjunction (logical multiplication) is true if and only if when all statements are true.

At least one of these disjunctions x*y*¬z; x*y*z; x*¬y*¬z will be true only if x=1.

Thus, the variable X corresponds to the column with variable 2 (column 2).

Let y- variable 1, z- prem.3. Then, in the first case x*¬y*¬z will be true in the second case x*y*¬z, and in the third x*y*z.

Answer: yxz

The symbol F denotes one of the following logical expressions from three arguments: X, Y, Z. A fragment of the truth table of the expression F is given (see the table on the right). Which expression matches F?

1) X ∧ Y ∧ Z 2) ¬X ∨ Y ∨¬Z 3) X ∧ Y ∨ Z 4) X ∨ Y ∧ ¬Z

Solution:

1) X ∧ Y ∧ Z = 1.0.1 = 0 (does not match on 2nd line)

2) ¬X ∨ Y ∨¬Z = ¬0 ∨ 0 ∨ ¬0 = 1+0+1 = 1 (does not match on the 1st line)

3) X ∧ Y ∨ Z = 0.1+0 = 0 (does not match on the 3rd line)

4) X ∨ Y ∧ ¬Z (corresponds to F)

X ∨ Y ∧ ¬Z = 0 ∨ 0 ∧ ¬0 = 0+0.1 = 0

X ∨ Y ∧ ¬Z = 1 ∨ 0 ∧ ¬1 = 1+0.0 = 1

X ∨ Y ∧ ¬Z = 0 ∨ 1 ∧ ¬0 = 0+1.1 = 1

Answer: 4

Given a fragment of the truth table of the expression F. Which expression corresponds to F?

1) (A → ¬B) ∨ C 2) (¬A ∨ B) ∧ C 3) (A ∧ B) → C 4) (A ∨ B) → C

Solution:

1) (A → ¬B) ∨ C = (1 → ¬0) ∨ 0 = (1 → 1) + 0 = 1 + 0 = 1 (does not match on 2nd line)

2) (¬A ∨ B) ∧ C = (¬1 ∨ 0) ∧ 1 = (0+0).1 = 0 (does not match on the 3rd line)

3) (A ∧ B) → C = (1 ∧ 0) → 0 = 0 → 0 = 1 (does not match on 2nd line)

4) (A ∨ B) → C (corresponds to F)

(A ∨ B) → C = (0 ∨ 1) → 1 = 1

(A ∨ B) → C = (1 ∨ 0) → 0 = 0

(A ∨ B) → C = (1 ∨ 0) → 1 = 1

Answer: 4

A logical expression is given that depends on 6 logical variables:

X1 ∨ ¬X2 ∨ X3 ∨ ¬X4 ∨ X5 ∨ X6

How many different sets of variable values ​​are there for which the expression is true?

1) 1 2) 2 3) 63 4) 64

Solution:

False expression only in 1 case: X1=0, X2=1, X3=0, X4=1, X5=0, X6=0

X1 ∨ ¬X2 ∨ X3 ∨ ¬X4 ∨ X5 ∨ X6 = 0 ∨ ¬1 ∨ 0 ∨ ¬1 ∨ 0 ∨ 0 = 0

There are 2 6 =64 options in total, which means true

Answer: 63

A fragment of the truth table of the expression F is given.

Which expression matches F?

1) x1 ∨ x2 ∨ ¬x3 ∨ x4 ∨ ¬x5 ∨ x6 ∨ ¬x7
2) x1 ∨ ¬x2 ∨ x3 ∨ ¬x4 ∨ ¬x5 ∨ x6 ∨ x7
3) x1 ∧ ¬x2 ∧ x3 ∧ ¬x4 ∧ x5 ∧ ¬x6 ∧ x7
4) x1 ∧ x2 ∧ ¬x3 ∧ x4 ∧ ¬x5 ∧ x6 ∧ ¬x7

Solution:

1) x1 ∨ x2 ∨ ¬x3 ∨ x4 ∨ ¬x5 ∨ x6 ∨ ¬x7 = 0 + 1 + … = 1 (does not match on the 1st line)

2) x1 ∨ ¬x2 ∨ x3 ∨ ¬x4 ∨ ¬x5 ∨ x6 ∨ x7 = 0 + 0 + 0 + 0 + 0 + 1 + 0 = 1 (does not match on the 1st line)

3) x1 ∧ ¬x2 ∧ x3 ∧ ¬x4 ∧ x5 ∧ ¬x6 ∧ x7 = 1.0. ...= 0 (does not match on 2nd line)

4) x1 ∧ x2 ∧ ¬x3 ∧ x4 ∧ ¬x5 ∧ x6 ∧ ¬x7 (corresponds to F)

x1 ∧ x2 ∧ ¬x3 ∧ x4 ∧ ¬x5 ∧ x6 ∧ ¬x7 = 1.1.1.1.1.1.1 = 1

x1 ∧ x2 ∧ ¬x3 ∧ x4 ∧ ¬x5 ∧ x6 ∧ ¬x7 = 0. … = 0

Answer: 4

What expression can F be?

1) x1 ∧ ¬x2 ∧ x3 ∧ ¬x4 ∧ x5 ∧ x6 ∧ ¬x7 ∧ ¬x8
2) ¬x1 ∨ x2 ∨ x3 ∨ ¬x4 ∨ ¬x5 ∨ ¬x6 ∨ ¬x7 ∨ x8
3) ¬x1 ∧ x2 ∧ ¬x3 ∧ x4 ∧ x5 ∧ ¬x6 ∧ ¬x7 ∧ ¬x8
4) ¬x1 ∨ ¬x2 ∨ ¬x3 ∨ ¬x4 ∨ ¬x5 ∨ ¬x6 ∨ ¬x7 ∨ ¬x8

Solution:

1) x1 ∧ ¬x2 ∧ x3 ∧ ¬x4 ∧ x5 ∧ x6 ∧ ¬x7 ∧ ¬x8 = x1 . ¬x2. 0 . ... = 0 (does not match on 1st line)

2) ¬x1 ∨ x2 ∨ x3 ∨ ¬x4 ∨ ¬x5 ∨ ¬x6 ∨ ¬x7 ∨ x8 (corresponds to F)

3) ¬x1 ∧ x2 ∧ ¬x3 ∧ x4 ∧ x5 ∧ ¬x6 ∧ ¬x7 ∧ ¬x8 = … ¬x7 ∧ ¬x8 = … ¬1 ∧ ¬x8 = … 0 ∧ ¬x8 = 0 (does not match on 1- th line)

4) ¬x1 ∨ ¬x2 ∨ ¬x3 ∨ ¬x4 ∨ ¬x5 ∨ ¬x6 ∨ ¬x7 ∨ ¬x8 = ¬x1 ∨ ¬x2 ∨ ¬x3 … = ¬1 ∨ ¬x2 ∨ ¬0 .. = 1 (not matches on the 2nd line)

Answer: 2

Given is a fragment of the truth table for the expression F:

Find the minimum possible number of different rows in the complete truth table of this expression in which the value x5 matches F.

Solution:

Minimum possible number of distinct rows in which x5 matches F = 4

Answer: 4

Given is a fragment of the truth table for the expression F:

Find the maximum possible number of distinct rows in the complete truth table of this expression in which the value x6 does not coincide with F.

Solution:

Maximum possible number = 2 8 = 256

The maximum possible number of different rows in which the value x6 does not match F = 256 – 5 = 251

Answer: 251

Given is a fragment of the truth table for the expression F:

Find the maximum possible number of different rows of the complete truth table of this expression in which the value ¬x5 ∨ x1 coincides with F.

Solution:

1+0=1 – does not match F

0+0=0 – does not match F

0+0=0 – does not match F

0+1=1 – coincides with F

1+0=1 – coincides with F

2 7 = 128 – 3 = 125

Answer: 125

Each Boolean expression A and B depends on the same set of 6 variables. In the truth tables, each of these expressions has exactly 4 units in the value column. What is the minimum possible number of ones in the value column of the truth table of the expression A ∨ B?

Solution:

Answer: 4

Each Boolean expression A and B depends on the same set of 7 variables. In the truth tables, each of these expressions has exactly 4 units in the value column. What is the maximum possible number of ones in the value column of the truth table of the expression A ∨ B?

Solution:

Answer: 8

Each Boolean expression A and B depends on the same set of 8 variables. In the truth tables, each of these expressions has exactly 5 units in the value column. What is the minimum possible number of zeros in the value column of the truth table of the expression A ∧ B?

Solution:

2 8 = 256 – 5 = 251

Answer: 251

Each Boolean expression A and B depends on the same set of 8 variables. In the truth tables, each of these expressions has exactly 6 units in the value column. What is the maximum possible number of zeros in the value column of the truth table of the expression A ∧ B?

Solution:

Answer: 256

The Boolean expressions A and B each depend on the same set of 5 variables. There are no matching rows in the truth tables of both expressions. How many ones will be contained in the value column of the truth table of the expression A ∧ B?

Solution:

There are no matching rows in the truth tables of both expressions.

Answer: 0

The Boolean expressions A and B each depend on the same set of 6 variables. There are no matching rows in the truth tables of both expressions. How many ones will be contained in the value column of the truth table of the expression A ∨ B?

Solution:

Answer: 64

Each of the Boolean expressions A and B depends on the same set of 7 variables. There are no matching rows in the truth tables of both expressions. What is the maximum possible number of zeros in the value column of the truth table of the expression ¬A ∨ B?

Solution:

A=1,B=0 => ¬0 ∨ 0 = 0 + 0 = 0

Answer: 128

Each of the Boolean expressions F and G contains 7 variables. There are exactly 8 in the truth tables of expressions F and G identical lines, and in exactly 5 of them there is a 1 in the value column. How many rows of the truth table for the expression F ∨ G contain a 1 in the value column?

Solution:

There are exactly 8 identical rows, and exactly 5 of them have a 1 in the value column.

This means that exactly 3 of them have a 0 in the value column.

Answer: 125

The logical function F is given by the expression (a ∧ ¬c) ∨ (¬b ∧ ¬c). Determine which column of the truth table of the function F corresponds to each of the variables a, b, c.

In your answer, write the letters a, b, c in the order in which their corresponding columns appear.

Solution:

(a . ¬c) + (¬b . ¬c)

When c is 1, F is zero so the last column is c.

To determine the first and second columns, we can use the values ​​from the 3rd row.

(a . 1) + (¬b . 1) = 0

Answer: ABC

The logical function F is given by the expression (a ∧ c)∨ (¬a ∧ (b ∨ ¬c)). Determine which column of the truth table of the function F corresponds to each of the variables a, b, c.

Based on the fact that when a=0 and c=0, then F=0, and the data from the second row, we can conclude that the third column contains b.

Answer: cab

The logical function F is given by x ∧ (¬y ∧ z ∧ ¬w ∨ y ∧ ¬z). The figure shows a fragment of the truth table of the function F, containing all sets of arguments for which the function F is true. Determine which column of the truth table of the function F corresponds to each of the variables x, y, z, w.

In your answer, write the letters x, y, z, w in the order in which their corresponding columns appear.

Solution:

x ∧ (¬y ∧ z ∧ ¬w ∨ y ∧ ¬z)

x. (¬y . z . ¬w . y . ¬z)

Based on the fact that at x=0, then F=0, we can conclude that the second column contains x.

Answer: wxzy