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The meaning of expressions in the mathematical language

In our everyday life we often need to interpret a given text precisely.
Even the slightest difference in the conjunctions might give a sentence with a completely different meaning.

Example 1

We omitted the expressions “at least” and “at most” from the leaflet of a travel
agency. Replace the missing expressions and interpret the sentences.
“.......... one piece of luggage with a weight of .......... \latex{15\, kg} can be checked in.
One piece of carry-on luggage with a weight of \latex{10\, kg} .......... can be taken
on board of the aeroplane. You should check-in .......... one hour prior
to the take-off of the aeroplane. You should have enough ‘pocket money’
on you for the travel, .......... \latex{30\, euros} per day.”

Solution

“At most one piece of luggage with a weight of at most \latex{15\, kg} can be checked in.” i.e. the pieces of luggage that can be checked in (d) is \latex{1} or less:
\latex{d} ≤ \latex{1}, and its weight (m₁) is \latex{15\, kg} or less: m₁ ≤ \latex{15\, kg}.

“One piece of carry-on luggage with a weight of at most \latex{10\, kg} can be taken on board of the aeroplane.” i.e. the weight of the carry-on luggage (m₂) can be \latex{10\, kg} or less: m₂ ≤ \latex{10\, kg}.

“You should check-in at least one hour prior to the take-off of the aeroplane.” i.e. the time between the check-in and the take-off of the aeroplane (t) is \latex{1} hour or more: t ≥ \latex{1\, h}.

“You should have enough pocket-money on you for the travel, at least \latex{30\, euros} per day.” i.e. the daily pocket-money (k) should be \latex{30\, euros} or more: k ≥ \latex{30\, euros}.

Example 2

We collected the number of boys (sons) and girls (daughters) of five families in a table. Based on it, list the families where:

all children are girls,
there is a boy,
it is not true that all children are girls,
it is not true that there is a boy.

Number of
girlsFamilyNumber of
boysEABCD2\latex{0}\latex{1}\latex{2}\latex{1}\latex{0}\latex{4}\latex{1}\latex{0}\latex{3}

Solution

All children are girls = there are no boys: in family A and E.
There is a boy: in family B, C and D.
It is not true that all children are girls: in family B, C and D.
It is not true that there is a boy: in family A and E.

We can realise that the statements of part a) and part d), as well as the statements of part b) and part c) are valid for the same families. Indeed, the following statements have the same meaning:

d) and a) are the negation of the proposition “ There is a boy.” :

It is not true that there is a boy. = All children are not boys. = All children are girls.

c) and b) are the negation of the proposition “All children are girls.” :

It is not true that all children are girls. = There is a child who is not a girl. = There is a boy.

If a proposition is true, then its negation is false, and if a proposition is false, then its negation is true.

The negation of the negation of a proposition is true exactly when the original proposition is true.

Example 3

For which pictures are the propositions true?

\latex{4}\latex{1}\latex{2}\latex{3}

It is raining and the wind is blowing.
It is raining or the wind is blowing.
It is not raining and the wind is not blowing.
It is not raining or the wind is not blowing.

Solution (a)

In picture \latex{1} It is raining and the wind is blowing. The proposition is true.
In picture \latex{2} it is not raining, in picture \latex{3} the wind is not blowing, in picture \latex{4} it is neither raining nor is the wind blowing, so the proposition is false for picture \latex{2}, \latex{3} and \latex{4}.

Solution (b)

In picture \latex{1} it is raining and the wind is blowing too, in picture \latex{2} the wind is blowing, in picture \latex{3} it is raining, so for these it is true that It is raining or the wind is blowing.

In picture \latex{4} it is neither raining nor is the wind blowing; the proposition is false for this picture.

Solution (c)

The proposition It is not raining and the wind is not blowing is true for picture \latex{4}, it is false for the other pictures.

We can see that this proposition is exactly the negation of the proposition
It is raining or the wind is blowing:

It is not true that it is raining or the wind is blowing. =
= It is not raining and the wind is not blowing.

Solution (d)

In picture \latex{2} it is not raining, in picture \latex{3} the wind is not blowing, in picture \latex{4} it is neither raining nor is the wind blowing, so for these it is true that It is not raining or the wind is not blowing.

In picture \latex{1} it is raining and the wind is blowing too; the proposition is false for this picture. This proposition is exactly the negation of proposition It is raining and the wind is blowing:

It is not true that it is raining and the wind is blowing. =
= It is not raining or the wind is not blowing.

Example 4

For which pictures are the propositions true?

\latex{4}\latex{1}\latex{2}\latex{3}

If it is noon then the sun is shining.
If the sun is shining then it is noon.
If it is not noon then the sun is not shining.
It is noon and the sun is not shining.

Solution (a)

In picture \latex{1} it is noon and the sun is shining, the proposition is true. In picture \latex{2} and \latex{4} the hypothesis is not fulfilled as it is not noon. Therefore whether the sun is shining, or it is not, the proposition If it is noon then the sun is shining is true. In picture \latex{3} it is noon, but the sun is not shining, proposition If it is noon then the sun is shining is false.

Solution (b)

Similarly to the train of thought in case a) the proposition If the sun is shining then it is noon is true for picture \latex{1}, \latex{3} and \latex{4}, it is false for picture \latex{2}.

Solution (c)

Proposition If it is not noon then the sun is not shining is true for picture \latex{1}, \latex{3} and \latex{4}, it is false for picture \latex{2}. The proposition is true exactly when the proposition If the sun is shining then it is noon is true, i.e. their meaning is the same.

Solution (d)

Proposition It is noon and the sun is not shining is true for picture \latex{3}, it is false for picture \latex{1}, \latex{2} and \latex{4}. This is exactly the negation of proposition a).

⯁ ⯁ ⯁

Statements a) and b) are each other's converse.

a) If it is noon, then the sun is shining.

b) If the sun is shining, then it is noon.

The negation of statement a) is statement d).

a) If it is noon, then the sun is shining.

d) It is noon and the sun is not shining.

⯁ ⯁ ⯁

In a computer game the machine generates sets A and B using the elements
of a logic set based on the characteristics of the elements. (Figure 1)

CHARACTERISTICS:largesmallfullredcircletrianglesquarewith holeyellowbluegreen

The elements of the logic set:Figure 1

The player has to find out these characteristics based on the machine placing the elements into the corresponding set part when the player clicks on an element of the logic set. The following is an example of a game:

The options left after the answerQuestionThe answer of the
machineA: triangle

B: greenA: small, full, yellow, blue, green,
triangle, square

B: small, full, yellow, blue, green,
triangle, squareA: small, full, blue, triangle

B: yellow, green, squareA: blue, triangle

B: green, square

After a few answers we might have an assumption about the characteristics which define the sets. When proving the assumption all other options should be excluded.

Exercises

{{exercise_number}}. Negate the following propositions.

There is a person who speaks at least \latex{15} languages.
All people have read at least one book from J. K. Rowling.
There is a beetle that is an insect.
Every triangle has an angle which is \latex{90º} at most.

{{exercise_number}}. How many books does Andy have if out of the following three statements exactly one is true?

Leslie: Andy has more than \latex{1,000} books.
Martin: No, he has less than \latex{1,000} books.
Tony: He has one book at least.

{{exercise_number}}. How many natural numbers not greater than \latex{20} are there for which the following statement is true?

Divisible by \latex{2} and \latex{3}.

Divisible by \latex{2} or \latex{3}.

Not divisible by either \latex{2} or \latex{3}.

Not divisible by \latex{2} or not divisible by \latex{3}.

{{exercise_number}}. Give the converse of statement If it is a bird, then it has wings, then give the negation of both the statement and its converse.

{{exercise_number}}. Write down the following statements and their converses, and decide whether they are true or false.

If number n is divisible by \latex{4}, then the number constituted from its last two digits is also divisible by \latex{4}.
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
If all faces of a body are squares, then the body is a cube.
If the absolute value of a number is greater than \latex{3}, then the number is also greater than \latex{3}.

{{exercise_number}}. Fill in the table with letters P, A, L, E so that each letter appears in every row, in every column and in every \latex{2\times2} square surrounded by thicker border.

LPEA

Puzzle

On a train Peter Williams, Terence Smith and Gabriel Jones are the conductor, the controller and the train operator, but not necessarily in this order. There are also three businessmen travelling on the train with the same surnames: Caleb Williams, Sigismund Smith and Edmund Jones.

\latex{1}. Sigismund Smith lives in Manchester.

\latex{2}. The controller lives half-way between Manchester and Birmingham.

\latex{3}. Edmund Jones earns exactly 40 thousand pounds a year.

\latex{4}. The next door neighbour of the controller is one of the passengers, and he earns exactly three times as much as the controller.

\latex{5}. Peter Williams defeats the conductor in wrestling.

\latex{6}. The passenger who has the same surname as the controller lives in Birmingham.

What is the name of the train operator?

Mathematics 9.

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