Quantifiers
(a)
Please see below.
I cannot work out why one is correct.
If $x < 0$, then there's no value $y \in \mathbb{R}$ so that $y^2 = x$.
(b) If I have $\exists$ followed by $\forall$, then does imply that there exists exactly one value applicable for all values in the second $\forall$ quantifier?
Say I have the natural numbers as my domain and:
$(\exists n)(\forall m) 2m=n$
Does it say there's exactly one number $n$, such that for all numbers $m$, $2m=n$? I'm a bit confused here.
Here is the relevant part to (a):
The order of the quantifiers counts.
\begin{align}&(\exists y \in \mathbb{R})(\forall x \in \mathbb{R})x=y^2\\&(\forall x \in \mathbb{R})(\exists y \in \mathbb{R})x=y^2\end{align} The first is false and the second is true.
Edited 20160608:
I would like to add another problem I have problems with:
.
The textbook gives TRUE for problem 2.
But, how can I find one value n, that holds true (for all) m?
Doesn't is say that there exists only one value that satisfies all m?
Thanks for your help!
It always helps to read it in English first and understand it.
(a) The first says that $(\exists y \in \mathbb{R})(\forall x \in \mathbb{R})x=y^2$. In English this says that there exists an element $y$ in $\mathbb{R}$ such that for all $x \in \mathbb{R}$, $x = y^2$. This is clearly false because there is no element $y \in \mathbb{R}$ such that every real number is equal to the square of it!
The second says that $(\forall x \in \mathbb{R})(\exists y \in \mathbb{R})x=y^2$. In English this says that for all $x \in \mathbb{R}$, there exists an element $y \in \mathbb{R}$ such that $x=y^2$. You are absolutely correct that it is not true for $x<0$, so this is probably a mistake in the text. If we look at $x \geq 0$ however, it is true, and this is probably what the text meant.
(b) Yes - almost. You say that "If I have $\exists$ followed by $\forall$, then does this imply that there exists $\color{red}{\text{exactly one}}$ value applicable for all values in the second $\forall$ quantifier?"
There does not need to be exactly one, we can have as many as we like. Related is $\exists!$, which means that there exists a $\color{red}{\text{unique}}$ element, i.e. exactly one.
To answer your question "if $\exists \forall$, does it imply that I need to find exactly one value from many, that fits the $\forall$? That's causing my confusion. Or I should say, the $\exists$ value is fixed, once I have chosen one?"
You want to find a value from all of $\mathbb{R}$ that when fixed, you can apply the $\forall$ part of the statement. So in your case you want to find some real number $y$ such that every real number is equal to the square of it.