# Linear Algebra/Quantifiers

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Compare these two statements about natural numbers: \"there is an $x$
such that $x$ is divisible by $x^2$\" is true, while \"for all numbers
$x$, that $x$ is divisible by $x^2$\" is false.
We call the \"there is\" and \"for all\"
prefixes **quantifiers**.

### For all

The \"for all\" prefix is the **universal quantifier**, symbolized
$\forall$.

Venn diagrams aren\'t very helpful with quantifiers, but in a sense the
box we draw to border the diagram shows the universal quantifier since
it delineates the universe of possible members.

```{=html}
<center>
```
![](Linalg_venn_forall.png "Linalg_venn_forall.png"){width=""
height="200"}

```{=html}
</center>
```
To prove that a statement holds in all cases, we must show that it holds
in each case. Thus, to prove \"every number divisible by $p$ has its
square divisible by $p^2$\", take a single number of the form $pn$ and
square it $(pn)^2=p^2n^2$. This is a \"typical element\" or \"generic
element\" proof.

This kind of argument requires that we are careful to not assume
properties for that element other than those in the hypothesis--- for
instance, this type of wrong argument is a common mistake: \"if $n$ is
divisible by a prime, say $2$, so that $n=2k$ then $n^2=(2k)^2=4k^2$ and
the square of the number is divisible by the square of the prime\". That
is an argument about the case $p=2$, but it isn\'t a proof for general
$p$.

### There exists

We will also use the **existential quantifier**, symbolized $\exists$
and read \"there exists\".

As noted above, Venn diagrams are not much help with quantifiers, but a
picture of \"there is a number such that $P$\" would show both that
there can be more than one and that not all numbers need satisfy $P$.

```{=html}
<center>
```
![](Linalg_venn_thereexists.png "Linalg_venn_thereexists.png"){width=""
height="200"}

```{=html}
</center>
```
An existence proposition can be proved by producing something satisfying
the property: once, to settle the question of primality of $2^{2^5}+1$,
Euler produced its divisor $641$. But there are proofs showing that
something exists without saying how to find it; Euclid\'s argument given
in the next subsection shows there are infinitely many primes without
naming them. In general, while demonstrating existence is better than
nothing, giving an example is better, and an exhaustive list of all
instances is great. Still, mathematicians take what they can get.

Finally, along with \"Are there any?\" we often ask \"How many?\" That
is why the issue of uniqueness often arises in conjunction with
questions of existence. Many times the two arguments are simpler if
separated, so note that just as proving something exists does not show
it is unique, neither does proving something is unique show that it
exists. (Obviously \"the natural number with more factors than any
other\" would be unique, but in fact no such number exists.)
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