Editing Uniqueness quantification (section)

== Proving uniqueness ==

The most common technique to prove the unique existence of an object is to first prove the existence of the entity with the desired condition, and then to prove that any two such entities (say, ''<math>a</math>'' and ''<math>b</math>'') must be equal to each other (i.e.&nbsp;<math>a = b</math>).

For example, to show that the equation <math>x + 2 = 5</math> has exactly one solution, one would first start by establishing that at least one solution exists, namely 3; the proof of this part is simply the verification that the equation below holds:

:<math> 3 + 2 = 5. </math>

To establish the uniqueness of the solution, one would proceed by assuming that there are two solutions, namely ''<math>a</math>'' and ''<math>b</math>'', satisfying <math>x + 2 = 5</math>. That is,

:<math> a + 2 = 5\text{ and }b + 2 = 5. </math>

Then since equality is a [[transitive relation]], 

:<math> a + 2 = b + 2. </math>

Subtracting 2 from both sides then yields
:<math> a = b. </math>

which completes the proof that 3 is the unique solution of <math>x + 2 = 5</math>.

In general, both existence (there exists ''at least'' one object) and uniqueness (there exists ''at most'' one object) must be proven, in order to conclude that there exists exactly one object satisfying a said condition.

An alternative way to prove uniqueness is to prove that there exists an object <math>a</math> satisfying the condition, and then to prove that every object satisfying the condition must be equal to <math>a</math>.