(related to Part: Methods of Mathematical Proving)
In many cases in mathematics, we are given an equation (or some equations) which depend on some unknowns and we want to find a solution to this / these equation(s). Let denote our equation by $f(x)=0$ and the unknown by $x$. The problem of finding a solution $x$ can usually be split into two different key questions.
This question is known as the existence problem of finding a solution for $f(x)=0.$ We can solve the existence problem by the following strategies:
Usually, it is possible to apply this strategy in some very few special cases. The disadvantage of this strategy is that it requires a way to solve the equation(s) and that is usually not applicable for finding the solutions in generalized cases.
Sometimes, it is possible to find a logical argument for the existence of a solution, for instance, by contradiction. The disadvantage of this strategy is that does not supply us with an explicit solution.
If we manage to find an algorithm which needs only a finite number of steps to find a solution to $f(x)=0,$ then this algorithm also solves the above existence problem.
If mathematicians succeed to solve the existence problem by applying e.g. the above-mentioned strategies, then they say
"$f(x)=0$ has at least one solution."
This question is known as the uniqueness problem of finding a solution for $f(x)=0.$ Like for the existence problem, there are multiple strategies to tackle the uniqueness problem:
First, we assume that $x,y$ are both solutions, i.e. $f(x)=0$ and $f(y)=0,$ and that $x\neq y.$ From these assumptions, we try to conclude a contradiction. This implies that $x=y$ which is equivalent to the uniqueness of any existing solution.
In this strategy, we try to conclude that from $x\neq y$ it follows that $x$ and $y$ cannot be both solutions of $f.$ If we succeed, then it implies the uniqueness of any existing solution.
If mathematicians succeed to solve the uniqueness problem by applying e.g. the above-mentioned strategies, then they say
"$f(x)=0$ has at most one solution."
Usually, the existence and the uniqueness problems can be treated independently from each other.
