Formalizing statements in logic

$A \equiv \left\{ \text{each of the numbers }a, b\text{ is divisible by 7}\right\} , B \equiv \left\{ \text{the sum }a+b\text{ is divisible by 7}\right\}$;

$A \equiv \left\{ \text{the last digit of the number } a \text{ is even}\right\} , B \equiv \left\{ \text{the number } A \text{ is divisible by 4}\right\}$;

$A \equiv \left\{ \text{triangle } A_{1} B_{1} C_{1} \text{ is isosceles}\right\}$, $B \equiv \left\{ \text{two medians of triangle } A_{1} B_{1} C_{1} \text{ are equal to each other}\right\}$;

$A \equiv \left\{ \text{from segments whose lengths equal } a, b, c \text{ a triangle can be formed}\right\} , B \equiv \left\{ \text{the positive numbers } a, b, c \text{ satisfy the inequalities } a+b > c, b+c > a, c+a > b\right\}$.

$\left\{ x_{1} < M, x_{2} < M\right\} \Leftrightarrow \left\{ D \geqslant 0, x_{0} < M, af(M) > 0\right\}$;

$\left\{ x_{1} > M, x_{2} > M\right\} \Leftrightarrow \left\{ D \geqslant 0, x_{0} > M, af(M) > 0\right\}$;

$\left\{ x_{1} < M < x_{2}\right\} \Leftrightarrow \left\{ a f(M) < 0\right\}$;

$\left\{ K < x_{1} < M, K < x_{2} < M\right\} \Leftrightarrow \left\{ D \geqslant 0, K < x_{0} < M, f(K) f(M) > 0\right\}$

$\left\{ x_{1} < K < M < x_{2}\right\} \Leftrightarrow \left\{ a f(K) < 0, a f(M) < 0\right\}$.

is unbounded above;

is unbounded below.

the function is not increasing;

the function is not decreasing;

the function is not monotone.

the number $a$ is not an upper bound of the function;

the number $a$ is not a lower bound of the function.

the value $f\left(x_{0}\right), x_{0} \in X$, is not the largest;

the value $f\left(x_{0}\right), x_{0} \in X$, is not the smallest value of the function on the set $X$.

$\forall m \exists n: x_{m} \neq x_{n}$;

$\exists N \forall n \geqslant N: x_{n} < x_{N}$;

$\exists N_{1} \forall n \geqslant N_{1}: x_{N_{1}} > x_{n}$ and $\exists N_{2} \forall n \geqslant N_{2}: x_{N_{2}} < x_{n}$;

$\exists N \forall n > N \forall m > n$ : $x_{n} < x_{m}$;

$\forall n \exists m > n \exists k > n: x_{m} < x_{n} < x_{k}$.

the sequence $\left\{ x_{n}\right\}$ is not increasing;

the sequence $\left\{ x_{n}\right\}$ is not decreasing.

there exists a natural number $N$ such that for any $\varepsilon > 0$ and any $n \geqslant N$ the inequality $\left|x_{n}-a\right| < \varepsilon$ holds;

for any $\varepsilon > 0$ there exist natural numbers $N$ and $n \geqslant N$ such that $\left|x_{n}-a\right| < \varepsilon$?

$\exists \varepsilon > 0 \exists N \quad \exists n \geqslant N:\left|x_{n}\right| < \varepsilon$;

$\exists \varepsilon > 0 \exists N \quad \forall n \geqslant N:\left|x_{n}\right| < \varepsilon$;

$\exists \varepsilon > 0 \quad \forall N \quad \exists n \geqslant N:\left|x_{n}\right| < \varepsilon$

$\forall \varepsilon > 0 \exists N \exists n \geqslant N:\left|x_{n}\right| < \varepsilon$;

$\exists \varepsilon > 0 \forall N \quad \forall n \geqslant N:\left|x_{n}\right| < \varepsilon$;

$\forall \varepsilon > 0 \exists N \quad \forall n \geqslant N:\left|x_{n}\right| < \varepsilon$;

$\forall \varepsilon > 0 \exists N \quad \forall n \geqslant N:\left|x_{n}\right| < \varepsilon$;

$\forall \varepsilon > 0 \quad \forall N \quad \forall n \geqslant N:\left|x_{n}\right| < \varepsilon$.

$\lim_{n \rightarrow \infty } x_{n} = \infty$;

$\lim_{n \rightarrow \infty } x_{n} = +\infty$;

$\lim_{n \rightarrow \infty } x_{n} = -\infty$

$\lim_{n \rightarrow \infty } x_{n} = \infty$;

$\lim_{n \rightarrow \infty } x_{n} = +\infty$;

$\lim_{n \rightarrow \infty } x_{n} = -\infty$.

the sequence does not tend to $\infty$;

the sequence does not tend to $+\infty$;

the sequence does not tend to $\mathbf{\kappa }-\infty$.

$0,2^{(-1)^{n} n}$

$\frac{n \cos \pi n-1}{2 n}$,

$(-1)^{n}\left(1+\frac{1}{n}\right)^{n}$,

$\left(1+\frac{(-1)^{n}}{n}\right)^{n}$;

$\frac{1}{2^{2}}+\frac{2}{3^{2}}+\ldots +\frac{n}{(n+1)^{2}}$.

if record $B$ is correct, then $A$ is erroneous;

at least one record of the pair $B, C$ is correct, and at least one record of the pair $A, C$ is correct;

if $A$ is erroneous, then exactly one of the records $B$ or $C$ is correct (but not both). Express the administrator's knowledge as a formula of propositional logic. Can he conclude that record $B$ is erroneous? Can it be reliably asserted that the erroneous record is unique?

if $z<0$, then $x<0$ or $y \geqslant 0$;

$x \geqslant 0$ or $y<0$;

if $y<0$, then exactly one of the variables $x$ or $z$ is negative, but not both. Express Pyotr's knowledge as a formula of propositional logic. Can he optimize the program by replacing the given conditional statement with the assignment $\mathrm{x}=1$; or with the assignment $\mathrm{x}=2$;? If ``yes'', with which one?

Misha decided to do a small renovation in his apartment and either put up new wallpaper or repaint the floor. Along with the wallpaper he will need to replace the laminate flooring, change the door, and replace the window's double-glazed unit. After repainting the floor he needs to install a stretch ceiling, a new chandelier, and new furniture. If the laminate is replaced, new furniture will also have to be bought. Old wallpaper cannot be left with old furniture, and old furniture cannot be left with an old chandelier.

If it is frosty, it will be sunny, or if it is overcast it will snow; and if it is not frosty, it will be sunny, or if it is overcast it will rain. In case of rain or snow it will be damp, and in case of rain it will also be muddy. If it is sunny, there will be neither dampness nor mud.

When Vova is home, his cat eats or sleeps, and when Vova is away, it also plays. If the cat eats, food needs to be bought. If the cat plays, new furniture or new wallpaper will have to be bought. For food one must go to the Everything for Cats'' pet shop, and for furniture or wallpaper to the Everything for the Home'' hardware store. Either shop requires taking a taxi. Vova will not take a taxi to the ``Everything for the Home'' shop.

$x$ is a grandson of $y$;

$x$ has at least two children;

$x$ and $y$ have one common child;

$x$ has an unmarried sister;

$x$ and $y$ are cousins;

$x$ is married, and only one of his sons is married.

$x$ is a number;

$f$ is a function;

$f$ is a constant function;

$f$ is a function identically equal to zero;

$x=0$;

$f$ is the identity function;

$x=1$;

$f$ is the exponential $e^{x}$;

$x+y=z-$ for numbers $x, y, z$;

$x y=z$ — for numbers $x, y, z$;

$x$ is a non-negative number;

the number $x$ is less than the number $y$;

the function $f$ takes all real values;

the function $f$ is one-to-one;

the function $f$ is sine;

the number $x$ equals $\pi$;

the number $x$ is a natural number.

$x$ is an even number;

$x$ and $y$ are relatively prime;

$z$ lies in the interval between $x$ and $y$;

$x$ is the greatest common divisor of $y$ and $z$;

$x$ is the least common multiple of all numbers in the interval $[y ; z)$.

$x, y$, and $z$ form a triangle;

$x$ is the midpoint of the segment $y z$;

the ray $x y$ is the bisector of the angle $\angle z x u$;

the angle $\angle x y z$ is a right angle;

the points $x, y, z, u$ lie on one circle;

the triangle $\triangle x y z$ is acute;

the segment $x y$ is shorter than $z u$;

the point $x$ lies inside the circle with center $y$ and a point $z$ lying on it.

Not all students study both analysis and history.

Only one student did not take the discrete mathematics exam.

Only one student passed all exams with 100 points.

The maximum score obtained in discrete mathematics exceeds the maximum score obtained in computer science.

There is a barber who shaves only those residents of the town who do not shave themselves.

There are politicians who can fool all the people some of the time; there are politicians who can fool some people all the time; but no one can fool all the people all the time.

in the relation Rooms, the set of attributes (EmployeeNumber, RoomNumber) is a key;

for every person in the relation Employees, their workplace is defined in the relation Rooms;

in the relation Rooms, the numbers of all rooms on the second floor are greater than 10 but less than 20, and the numbers of all rooms on the third floor are greater than 20.

$(\neg A \rightarrow (B \wedge C))$;

$((A \wedge B) \vee (C \wedge D))$;

$((A \wedge B) \vee (A \oplus C))$;

$((A \wedge \neg B) \vee (C \wedge D))$;

$((A \wedge B) \vee (A \wedge C))$;

$(A \wedge (B \oplus C))$?

equals 6;

is divisible by 4;

is divisible by 3;

is prime;

is composite;

is a Fibonacci number.

$x_{1}=x_{2}$;

$x_{1}<x_{2}$;

$x_{1} \leqslant x_{2}$.

if record $B$ is correct, then $A$ is erroneous;

at least one record of the pair $B, C$ is correct and at least one record of the pair $A, C$ is correct;

if $A$ is erroneous, then at least one of the records $B$ or $C$ is correct (but not both together). Describe the administrator's knowledge as a formula of propositional logic. Can he conclude that record $B$ is erroneous? Can it be reliably asserted that the erroneous record is unique?

if $z<0$, then $x<0$ or $y \geqslant 0$;

$x \geqslant 0$ or $y<0$;

if $y<0$, then at least one of the variables $x$ or $z$ is negative, but not both together.

Describe Peter's knowledge as a formula of propositional logic. Can he optimize the program by replacing the given conditional statement with the assignment $\mathrm{x}=1$; or with the assignment $\mathrm{x}=2$;? If "yes", then with which one?

Misha decided to do a small renovation in his apartment and either put up new wallpaper or paint the floor. Along with the wallpaper he will need to renew the laminate flooring, replace the doors, and replace the double-glazed window. After painting the floor he needs to install a stretch ceiling, a new chandelier, and furniture. If the laminate flooring is renewed, then new furniture will also have to be bought. The old wallpaper cannot be left with the old furniture, nor can the old furniture be left with the old chandelier.

If there is frost, then it will be sunny, or if the weather is overcast it will snow, and if there is no frost, then it will be sunny, or if the weather is overcast it will rain. In case of rain or snow it will be damp, and in case of rain it will also be muddy. If it is sunny, then there will be neither dampness nor mud.

When Vova is at home, his cat eats or sleeps, and if Vova is not there, it also plays. If the cat eats, cat food needs to be bought. If the cat plays, new furniture or new wallpaper will have to be bought. For food one needs to go to the pet shop "Everything for Cats", and for furniture or wallpaper — to the hardware store "Everything for the Home". A taxi is needed to get to either shop. Vova will not take a taxi to the "Everything for the Home" store.

if $B$ is the criminal, then $C$ is also the criminal;

someone from the pair $A, C$ is the criminal, but not both together;

if $C$ is not the criminal, then $A$ is not the criminal either.

Describe Sh. Holmes's knowledge as a formula of propositional logic and construct its truth table. Can he conclude that $C$ is the criminal? Can it be reliably asserted that the criminal acted alone?

if investments do not increase, then either government spending will increase or unemployment will arise;

if government spending does not increase, then taxes will decrease;

if taxes decrease and investments increase, then unemployment will not arise.

It turned out that at the present moment investments have not increased. Can the students conclude that government spending will increase?

Six men, $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}$, and F, are the only members eligible for the offices of President, Vice-President, and Secretary in Club NP. The following preferences of these men are known:

• A won’t be an officer unless E is President.

• B won’t serve if he outranks C.

• B won’t serve with F under any conditions.

• C won’t serve with both E and F.

• C won’t serve if F is President or B is Secretary.

• D won’t serve with C or E unless he outranks them.

• E won’t be Vice-President.

• E won’t be Secretary if D is an officer.

• E won’t serve with A unless F serves too.

• F won’t serve unless either he or C is President.

How can the three offices be filled satisfying all above conditions?

Police arrested four men as the suspects of a murder case. They made the following statements when questioned:

John: "Nick did it." Nick: "Bill did it." Dan: "I didn't do it." Bill: "Nick lied when he said I did it."

If exactly one of them is the real murderer and exactly one of the above statements is true, who is the murderer?

Brown, Jones, and Smith are employed by the village of NP as fireman, policeman, and teacher, though not necessarily respectively. Someone in the village reported that:

• Brown and the teacher are neighbors.

• Jones and the teacher are neighbors.

• Both Brown and Smith are neighbors of the fireman.

• Both the policeman and the fireman are neighbors of Jones.

• The men are all neighbors.

However, the truth of the matter is that only two of these statements are true. Can you determine the job which each man holds?

Both the Smiths and the Taylors have two young sons under eleven. The names of the boys, whose ages rounded off to the nearest year are all different, are Arthur, Bert, Carl, and David. Taking the ages of the boys only to the nearest year, the following statements are true:

• Arthur is three years younger than his brother.

• Bert is the oldest.

• Carl is half as old as one of the Taylor boys.

• David is five years older than the younger Smith boy.

• The total ages of the boys in each family differ by the same amount today as they did five years ago.

How old is each boy, and what is each boy's family name?"