To prove that a real number x satisfying x² + x − 6 = 0 exists, a student names x = 2 and stops. What is missing from this proof?
ANothing — naming a specific value is sufficient for an existence proof
BA uniqueness argument showing no other value satisfies the equation
CVerification that x = 2 actually satisfies x² + x − 6 = 0
DA proof by contradiction ruling out all other candidates
Naming a witness is the right strategy for an existence proof, but naming alone is not sufficient — you must verify that the named object actually satisfies the predicate. The student should confirm: 2² + 2 − 6 = 0. ✓ Without this verification, you've only made a claim. The act of checking is the proof.
Question 2 Multiple Choice
To prove that the solution to Ax = b (for an invertible matrix A) is unique, which strategy is correct?
AShow that the formula x = A⁻¹b gives a specific answer — this establishes uniqueness
BAssume x and y are both solutions, then derive x = y using properties of A
CShow that A⁻¹ exists — invertibility alone implies the solution must be unique
DProve that no other value satisfies Ax = b by checking every possible vector
The standard uniqueness proof strategy is 'assume two, show they're equal': suppose Ax = b and Ay = b, then A(x − y) = 0. Since A is invertible, x − y = 0, so x = y. Option A (showing the formula gives a specific answer) establishes existence, not uniqueness — it finds at least one solution but doesn't prove there can't be others. The uniqueness proof must begin by assuming two solutions might exist and then deriving they are equal.
Question 3 True / False
A uniqueness proof should begin by assuming that there is only one solution to the problem being proved.
TTrue
FFalse
Answer: False
This is the key misconception about uniqueness proofs. You do not assume uniqueness — you prove it. The standard technique begins by assuming the opposite: suppose x and y are both solutions (possibly equal, possibly not). Then you derive x = y. If any two solutions must be equal, the solution is unique. Starting by assuming uniqueness would be circular; the proof works precisely because it makes no such assumption upfront.
Question 4 True / False
A proof that establishes existence automatically establishes uniqueness — if you can construct a specific object, it is expected to be the main one.
TTrue
FFalse
Answer: False
Existence and uniqueness are entirely independent. Existence says 'at least one solution exists'; uniqueness says 'at most one solution exists.' A construction can yield one of many possible solutions. For example, x² = 4 has two real solutions (x = 2 and x = −2) — existence is trivial, but uniqueness fails. You can construct x = 2 without that being the only solution. This is why existence-and-uniqueness proofs require both components separately.
Question 5 Short Answer
Why does mathematics care about proving both existence and uniqueness, rather than just finding a solution?
Think about your answer, then reveal below.
Model answer: Existence tells you the problem is not vacuous — there is something to find. Uniqueness tells you the solution is well-defined — it makes sense to speak of 'the' solution rather than 'a' solution. Together they justify computing a specific answer and trusting it is the only right one. Without uniqueness, a solution you compute might be one of many, and a different computation could yield a different, equally valid answer.
Consider Ax = b: if A is not invertible, there may be zero solutions (inconsistent) or infinitely many (underdetermined). Existence rules out the first case; uniqueness rules out the second. Only when both hold can you trust that 'solve Ax = b' has a single definitive answer. In physics, a differential equation modeling a physical system should have a unique solution given initial conditions — otherwise the theory fails to make unique predictions.