Quantifier

Every Hunan being has a certain tendency to generalize a thought, for example, if a person has political thoughts contrary to yours, then you are from that political wing. Mainly with the right/left political polarization, but that’s just an example.

To avoid controversy, we will present other examples:

• “All cats are yellow.”
• “There is at least one green alligator.”

These are examples of simple propositions. When we use connectives, problems begin to appear. For example:

• p: “all mice are black”;
• q: “all mice are white.”

White mouse	black mouse

Connectives

When using the connective “and” for the propositions above, we have that $p$ and $q$ ($p\wedge q$) represents that:

• all mice are black and white.
• Do mice have two colors?

Now if we use the connective “or”, we have that $p$ or $q$ ($p\vee q$) represents that:

• all mice are black or white.
• In this case the sentence is true if:
  • all the mice are black or all are white.
  • are all black and white at the same time.

The word todos is an example of a quantifier, there are other quantifiers such as:

• Some
• None
• Among others

Auxiliary connectors

There are other quantifiers that help the connectives mentioned above to work as conceptors, for example:

• There is $(\exist)$
• For all $(\forall)$

It is worth remembering that a mathematician is not a calculator, but a person who generalizes a thought. For example:

• The mathematician does not calculate $1 + 2 = 3$
• And yes, given $x$, $y$ two natural numbers, the sum whose result is 3 is given by:
  • $x$ can be 1 or 2, just as $y$ can be 1 or 2.
  • Suppose $x$ 1, this way we get $y$ equal to 2 or vice versa.
  • With this, we note that $x+y=y+x = 3$ regardless of whether $x=2$ or $x=1.$

Generalization of thinking

Therefore, in a more generic way of thinking we have to:

• $x$ is white
• Therefore, $x$ is any object, which can be animal, plant or any other object that we can attribute to $x$.

For now, we cannot say whether the proposition is true or not.

Now when using the auxiliary connective for todo, we can reformulate the proposition.

• For all $x$, $x$ is white.
• $\forall x, \quad x ~ \text{é branco}$
• Given $x$ represented by the mouse, we can verify that it is the same as
  • “All mice are white.”

It is worth remembering that for this example it is false. The idea is not to talk about whether the proposition is false or true, but to generalize the thought. The $x$ It can be any object, which could be assigned any adjective.

Now notice that when we use the connective exist $(\exist)$, we can reformulate the proposition.

• There is $x$, such that for some $x$ is white.
• $\exist x, ~ \text{tal que, para algum } x ~ \text{é branco}$
• Given $x$ represented by the mouse, we can verify that it is the same as
  • “There is at least one white mouse.”

This proposition may be true.

In mathematics, what does $x$ $(\forall x)$ represent for all?

The answer is that for any $x$ we take, the assigned proposition is true.

• Let $B(x)$ represent the object $x$ as white.
• Let $P(x)$ represent the object $x$ as black.
• We can reformulate the proposition as:
  • $\forall x, ~(B(x) \vee P(x))$, that is, for every (or for each) mouse, the mouse is white or the mouse is black.
  • which is different from $\forall x, ~(B(x) \wedge P(x))$, that is, for all (or for each) mouse, the mouse is white and black (spotted type).

Summary of thought:

Let $Q(x)$ be any sentence

• $(\forall x, Q(x) )$ means that for every $x$, the statement $Q(x)$ is true.
• $(\exist x, Q(x))$ means that we have at least one value of $x$ such that $Q(x)$ is true.

The sentence $Q(x)$ can be composed as $(B(x) \vee P(x))$ or another definition.

Denial of sentence

A very important problem that can cause a lot of confusion is when we deny a sentence. For example, we take:

By denying this sentence, we have:

What does this sentence mean? The answer is that it is not true that every $x$ we take, the statement $Q(x)$ satisfies the property. Equivalent to:

Here we must be very, very careful, as this is different from:

This sentence means that for all values ​​of $x$ that we take, it does not satisfy the property $Q(x)$. In other words it is the same as:

In the next posts we will use these concepts to demonstrate results.