# Example: Existence and Uniqueness of Solutions

(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.

### Question 1: Does $f(x)=0$ has at least one solution?

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:

# Strategy 1: Finding an explicit solution

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.

# Strategy 2: Abstract argumentation

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.

# Strategy 3: Finding a procedure (algorithm) to find a 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."

### Question 2: Does $f(x)=0$ has at most 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:

# Strategy 1: Proof by contradiction

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.

# Strategy 2: Proof by contraposition.

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.

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### References

#### Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016
2. Knabner, P; Barth, W.: "Lineare Algebra - Grundlagen und Anwendungen", Springer Spektrum, 2013