Understanding Existential and Universal Quantifiers in Mathematical Statements

[scorpius1782]

Homework Statement

I'm given a series of statements and I have to decide which one is true for my problem.
Are any of these equivalent to $\exists ! xP(x)$?

$\exists x P(x)$^$\forall y(p(y)\implies y=x)$

$\exists x \forall y, x=y \implies p(y)$

$\exists x \forall y, p(y) \implies x=y$

And true or false?

$\forall y \in \mathbb{N}, \exists x \in \mathbb{N}, x \leq y$
$\exists x \in \mathbb{N}, \forall y \in \mathbb{N}, x \leq y$

Homework Equations

The Attempt at a Solution

Just reading them properly is my big problem.
So for $\exists x P(x)$^$\forall y(P(y)\implies y=x)$

it is saying there exists an 'x' satisfying P(x) and all 'y' then if P(y)=x then y=x.

I bolded the part I'm not sure about. It is the inclusion of the second element that throws me. It is the same for the true or false questions:

$\exists x \in \mathbb{N}, \forall y \in \mathbb{N}, x \leq y$

There exists an 'x' in the natural numbers that for all 'y' in the natural numbers $x \leq y$

I would say this is true because x could be the same as y.

Thanks for any help.

 

[micromass]

Homework Statement

I'm given a series of statements and I have to decide which one is true for my problem.
Are any of these equivalent to $\exists ! xP(x)$?

$\exists x P(x)$^$\forall y(p(y)\implies y=x)$

$\exists x \forall y, x=y \implies p(y)$

$\exists x \forall y, p(y) \implies x=y$

And true or false?

$\forall y \in \mathbb{N}, \exists x in \mathbb{N}, x \leq y$
$\exists x \in \mathbb{N}, \forall y \in \mathbb{N}, x \leq y$

Homework Equations

The Attempt at a Solution

Just reading them properly is my big problem.
So for $\exists x P(x)$^$\forall y(P(y)\implies y=x)$

it is saying there exists an 'x' satisfying P(x) and all 'y' then if P(y)=x then y=x.

But $P(y)$ is a property of $y$ which can be true or false. So $P(y)=x$ makes no sense at all.
For example $P(y)$ can be "$y$ is colored green" or "$y$ is larger than $2$".
I don't know what $P(y)=x$ is supposed to mean.

It is the same for the true or false questions:

$\exists x \in \mathbb{N}, \forall y \in \mathbb{N}, x \leq y$

There exists an 'x' in the natural numbers that for all 'y' in the natural numbers $x \leq y$

I would say this is true because x could be the same as y.

If you claim this is a true statement, then what is the $x$ that makes the statement true?

 

[scorpius1782]

If you claim this is a true statement, then what is the $x$ that makes the statement true?

It would have to be zero, correct? It is less than or equal to all the other elements in the natural numbers. I'm confused about the order of the statements for the both of the true or false statements. I have no idea if order matters because I'm not sure if I'm able to put it into words correctly.

But $P(y)$ is a property of $y$ which can be true or false. So $P(y)=x$ makes no sense at all.
For example $P(y)$ can be "$y$ is colored green" or "$y$ is larger than $2$".
I don't know what $P(y)=x$ is supposed to mean.

I honestly don't know either. I just don't have examples of statements put into words so that I can properly read these. The only ones in my book are really simple ones with 1 variable in 1 set. I'm scouring the internet trying to find complicated examples that I can learn from but not much luck yet.

 

[micromass]

It would have to be zero, correct? It is less than or equal to all the other elements in the natural numbers.

Yes.

I'm confused about the order of the statements for the both of the true or false statements. I have no idea if order matters because I'm not sure if I'm able to put it into words correctly.

You mean the order of the quantifiers, right? Yes, that order is extremely important. The statement

$$\exists x\in \mathbb{N}: ~\forall y\in \mathbb{N}:~x\leq y$$

Means that there exists some natural number $x$ such that all natural numbers are greater or equal than $x$. This is true, and we can take $x=0$.

However,

$$\forall x\in \mathbb{N}:~\exists y\in \mathbb{N}:~x\leq y$$

means that for each natural number $x$, there is some greater natural number $y$. This is also true. For example, if $x$ is a natural number, then we can take $y=x+1$ (or even $y=x$). This is greater than $x$.

I honestly don't know either. I just don't have examples of statements put into words so that I can properly read these. The only ones in my book are really simple ones with 1 variable in 1 set. I'm scouring the internet trying to find complicated examples that I can learn from but not much luck yet.

Well, something like $P(y)=x$ makes no sense at all.

You can try imagine that $y$ is a car and that $P(y)$ is "the color of $y$ is green". In that sense, the statement

$$\exists x:~P(x)~\wedge~\forall y:~(P(y)~\Rightarrow~x=y)$$

means that

There exists a car $x$ such that $x$ is greeen, and for all other cars $y$ holds that if $y$ is green then $x=y$.

Or simplified:

There exists a green car $x$ and if some car $y$ is green, then $x=y$.

You see that this asserts that there is a unique green car.

Can you interpret the other $2$ statements by using green cars?

 

[scorpius1782]

$\exists x \forall y, x=y \implies p(y)$

There exists a car, x, and all other cars, y, that if x=y then y is green. So, x must also be green.

$\exists x \forall y, p(y) \implies x=y$

There exists a car, x, and all other cars, y, that if y is green then x=y. So, x must also be green.

The second one implies, to me, that several cars could be green. The first, seems like it is saying that if a car is green then it must be 'x.' Meaning there is just one green car.

 

[micromass]

$\exists x \forall y, x=y \implies p(y)$

There exists a car, x, and all other cars, y, that if x=y then y is green. So, x must also be green.

The "and" that I bolded above is incorrect. If you should read an "and", then it would have said $\wedge$.

How you should read it is

There is some car $x$ such that if $y$ is any car such that $x=y$, then $y$ is green.

Thus

There is some car $x$ such that any car equal to $x$ is green.

Thus this just says that

There exists a green car.

This statement does not say green cars are unique. Do you see that?

$\exists x \forall y, p(y) \implies x=y$

There exists a car, x, and all other cars, y, that if y is green then x=y. So, x must also be green.

Same here, you shouldn't have put in the "and". What this statement says is

There exists some car $x$ such that if $y$ is any green car, then $x=y$.

Thus

There exists some car $x$ such that all green cars equal $x$.

Thus

There is at most one green car.

This statement does not say that there is a green car. Do you see that?

 

[scorpius1782]

Okay, yeah I understand those now. That's what I was really having trouble with. That is how to interpret the function of the second variable in the beginning. This makes a lot more sense to me now.