Proving Uniqueness
Proving uniqueness turns out to be mostly easier than that of existence or expressibility. The most common technique to proving uniqueness is to assume there exists two quantities (say, a and b) that satisfies the condition given, and then logically deducing their equality, i.e. a = b.
As a simple high school example, to show x + 2 = 5 has only one solution, we assume there are two solutions first, namely, a and b, satisfying x + 2 = 5. Thus
By transitivity of equality,
By cancellation,
This simple example shows how a proof of uniqueness is done, the end result being the equality of the two quantities that satisfy the condition. We must say, however, that existence/expressibility must be proven before uniqueness, or else we cannot even assume the existence of those two quantities to begin with.
Read more about this topic: Uniqueness Quantification
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