Definition
Uniqueness
Uniqueness means that exactly one object or value satisfies all the stated conditions.
What it means
Finding one answer shows that an answer exists. It does not yet show that the answer is the only one. To establish uniqueness, check the full set of possibilities or use the conditions to rule out every possible second answer.
Remember it
Find one, then close every door to a second.
Example
Find a whole number greater than 20 and less than 30 that is divisible by both 3 and 4. Such a number must be a multiple of 12. The only multiple of 12 in that interval is 24, so 24 exists and no second value works. The solution is unique.
Watch for
Producing one valid answer proves existence, not uniqueness, unless all alternatives have also been ruled out.
Common mix-up
Unique means “one and only one,” not “unusual” or “interesting.”
Exact meaning
Uniqueness is the property that exactly one object or value satisfies the specified conditions. A uniqueness argument establishes that a solution exists and that no different second solution can satisfy those conditions.