Suppose that after sufficient work on a statement, we have managed to obtain the following information:

| $\mathrm{A}$ | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| $\mathrm{B}$ | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| $\mathrm{C}$ | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| $\mathrm{A} \Leftrightarrow \mathrm{B}$ | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |

And we are wondering if $\mathrm{A} \Rightarrow(\mathrm{B} \Leftrightarrow \mathrm{C})$?
It is necessary to add a line to our truth table that will precisely test the truth of what we propose to establish:

| $\mathrm{A}$ | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| ---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $\mathrm{B}$ | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| $\mathrm{C}$ | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| $\mathrm{A} \Leftrightarrow \mathrm{B}$ | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
| $\mathrm{A} \Rightarrow(\mathrm{B} \Leftrightarrow \mathrm{C})$ | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 |

The last line thus shows in which cases our proposition is true. We note that in two cases, our proposition is false, from which we conclude that:

$$
A \nRightarrow(B \Leftrightarrow C)
$$

## Quantifiers and Associated Symbols

Constructing a mathematical sentence requires quantifiers to express one's thoughts. To make the formulation less dependent on language and to shorten expressions, each quantifier has been associated with a symbol:

$\Delta$ The universal quantifier:

This quantifier allows for generalizations and can be expressed with the phrases "for all" or "whatever". It is associated with the symbol $\forall$. For example:

$$
\forall n \in \mathbb{N}
$$

which reads and means:

"For all $n$ natural numbers" or "whatever $n$ natural number"

$\Delta$ The existential quantifier:

This quantifier indicates the existence of an object with certain properties. It can be expressed with the phrase "there exists". It is associated with the symbol $\exists$. For example:

$$
\exists n / \forall k \in \mathbb{N}, n \leq k
$$

which reads:

"There exists $n$ such that for all $k$ natural numbers, $n \leq k$"

Of course, the true value of $n$ is known, it is $n=0$.

The negation of this quantifier also exists, it is "there does not exist", denoted by $\nexists$.

This quantifier has a slight variation to indicate that the object whose existence is indicated is also unique: it is denoted by $\exists!$ which allows, for example, to say:

$$
\forall x \in \mathbb{R} \backslash\{0\}, \exists!y \in \mathbb{R} \backslash\{0\} / x=\frac{1}{y}
$$

There are also practical notations that are not quantifiers but should be known. Here is their meaning:

$\triangleright$ Membership: "belongs to" is denoted by $\in$

An important point is to be aware that this notation links two objects of different status: for example, $a \in A$ expresses that $a$ is an element of $A$, on the left an element, on the right, the set to which it belongs.

Be careful, when starting to consider sets of sets, sets of sets of sets, it is necessary to verify the status of each one every time to avoid writing absurdities.

For example, we can write $n \in \mathbb{N}$, but we must not write $\mathbb{N} \in \mathbb{Z}$

The negation exists, and to say "$n$ does not belong to", we write $\notin$

## $\triangleright$ Inclusion:

This is the concept that should not be confused with membership. We say that $A$ is included in $B$ and we write $A \subset B$ to express that all elements of $A$ are also elements of $B$. $A$ and $B$ are therefore sets on the same hierarchical level.

For example, we have: $\mathbb{N} \subset \mathbb{Z}$

The notation "$n$ is not included in" exists, it is $\not \subset$
$\Delta$ Properties:

The expression "such that" is generally abbreviated in mathematics by the notation /

For example, we will write "let $n$ such that $2 n \leq k$" as:

let $n / 2 n \leq k$

## Constructing Logical Sentences

All these elements are useful to mathematicians for formulating logical sentences. Thus, chains of symbols are formed that serve as sentences. For example,

"For all natural number $n$, there exists an integer $k$ such that $n \leq k$" can be translated mathematically as:

$$
\forall n \in \mathbb{N}, \exists n \in \mathbb{N} / n \leq k
$$

Be careful, the order of the quantifiers is of paramount importance! We readily agree on the truth of the previous proposition; however, a careless student might write:

$$
\exists k \in \mathbb{N} / \forall n \in \mathbb{N}, n \leq k
$$

which is violently false since it means that all integers are less than the same number.

## $\triangleright$ Negation of Logical Sentences:

Like any sentence, a logical sentence can be put in its negative form. The simplest method to do this is to work not with the symbolic expression, but with words that make sense. There is nothing more cumbersome than elaborating an algorithmic method to negate a mathematical sentence. Therefore, one must use common sense, think about the meaning of the sentence, construct the negation in plain English, and then return to the mathematical language. Some exercises to practice: