What does an introductory-level proof of existence look like?
For example, I have done proofs involving proving uniqueness by assuming there are two arbitrary elements that have the properties I want, and then showing that they are actually the same element.
But let's say I want to prove that a solution exists at all (or that one definitely does not exist).
So, for $ax = b$, I could show that there is a unique $x$ by assuming there are two elements that would satisfy that equation, assuming the existence of a solution. How could I first prove that a solution may or may not even exist though?
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$\begingroup$One technique is just to write down the object and show it satisfies the properties you're looking for. So you would write down the number $\frac{b}{a}$ and show that this number solves the equation.
Existence proofs aren't always so direct, you might show that non-existence of an object produces a contradiction. You also might use the existence statement of a different theorem to prove existence of your object, for example you might note that the map $x \mapsto ax$ is surjective therefore $b$ is in the image. On a completely formal level this proves a solution exists without explicitly writing it down. It's a bit artificial in this case because we know which element maps to $b$ under this map, but there are more complicated situations where you can't transform this into a direct proof so easily, or even at all.
There are also some interesting proofs in statistics and combinatorics where one proves that an object exists by proving that if an object is drawn at random then there is a positive probability of it being an object of the type desired.
In general existence proofs are very situation dependent. There's not much one can say without a specific example in mind.
$\endgroup$ $\begingroup$In $\mathbb N$, the equation $ax=b$ does not have a solution in general. You can prove it by the counterexample
$$2x=1,$$
which cannot work because an even number cannot be odd.
In $\mathbb Q$, $ax=b$ has a solution by definition (except for $a=0$). Indeed,
$$\frac{a_n}{a_d}\frac{x_n}{x_d}=\frac{b_n}{b_d}\iff\frac{x_n}{x_d}=\frac{a_d}{a_n}\frac{b_n}{b_d}$$ by definition of multiplication and division and the associated group structure.
Existence/non-existence proofs are found case-by-case, there is no general rule. Sometimes you can prove existence by exhibiting a solution, and non-existence by contradiction.
$\endgroup$ $\begingroup$In his case it's easiest to prove existence exist by simply finding something that works.
$a(a^{-1}\cdot b)= (a\cdot a^{-1})\cdot b = e\cdot b = b$. (But I magically had to guess that $a^{-1}\cdot b$ would fit in.)
So $x = a^{-1}\cdot b$ is a solution. Then to prove it is unique
assume $ax = b$ and ..... solve for $x$.... $a^{-1}ax = a^{-1}b$ so $x = a^{-1}b$ is the only solution.
Now it's okay to say: We'll prove uniqueness first. Let's assume a solution $x$ exists so that $ax=b$ and so.... $x = a^{-1}b$ is the only solution. Now to prove a solution exists at all.... well, every step was bidirectional so we have shown that $x= a^{-1}b$ is a solution.
What (sort of) is not okay, depending on how fussy your instructor is, is to simply solve. $ax = b$ so $x = a^{-1}b$, is not really correct because you never stated you making an assumption that any solution exists, and in the process of solving you never stated that verifies the assumption.
It's subtle.... and very picayune. And in any course after the first few weeks it'd be assumed that whenever you are "solving" you are implying without explicitly stating that you are verifying existence and proving uniqueness simultaneously and silently implicitly. And it will be assumed that needs not be explicitly stated (it really doesn't).
But it is an subtle important concept one needs to be aware of.
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