Existence and Uniqueness Proofs

How do you prove that something exists — and only one such thing? The two halves of "exactly one" need two completely different arguments. Existence is shown by producing an object: build it, pick it, or argue (often non-constructively) that one must exist. Uniqueness is shown by the opposite move: suppose two objects satisfy the property and prove they must be the same one. The combined claim $\exists! x, P(x)$ — "there is exactly one $x$" — appears constantly in algebra (the identity element is unique), analysis (the limit of a convergent sequence is unique), and elsewhere. This section is short on prose because the moves are simple, but rigorous on bookkeeping because mixing the two halves is a classic mistake.

Constructive existence: exhibit the object

To prove $\exists x \in D, P(x)$ constructively:

1. Produce a specific $c \in D$.
2. Verify $P(c)$.
3. Conclude existence. $\blacksquare$

The witness $c$ can come from solving an equation, applying a known construction, or making a clever choice. To prove "there is a real $x$ with $x^3 = 8$," take $x = 2$; $2^3 = 8$. Done. The witness need not be unique — any object that satisfies $P$ suffices for existence.

Non-constructive existence: an object must exist

Sometimes you can prove existence without exhibiting the object explicitly. The classic example:

Theorem. There exist irrational $a, b$ such that $a^b$ is rational. Proof. Consider $\sqrt{2}^{\sqrt{2}}$. Either it is rational (take $a = b = \sqrt{2}$, both irrational, done) or it is irrational (take $a = \sqrt{2}^{\sqrt{2}}$ and $b = \sqrt{2}$; then $a^b = (\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = \sqrt{2}^2 = 2$, rational). Either way the existence claim holds, but the proof does not say which $a$ we should use. $\blacksquare$

Non-constructive proofs are logically airtight but provide no algorithm for finding the object. Constructive mathematics rejects them; classical mathematics accepts them. In undergraduate work, both are valid — just know which you have produced.

Uniqueness: suppose two, prove one

To prove there is at most one object satisfying $P$:

1. Suppose $a$ and $b$ both satisfy $P$.
2. Derive $a = b$.
3. Conclude uniqueness. $\blacksquare$

This is sometimes called the "suppose two, show one" strategy. The key is that the proof uses only that $a$ and $b$ satisfy $P$; no special features of either are invoked, so the conclusion $a = b$ applies to every pair of property-satisfying objects.

Classic example: the additive identity in $\mathbb{R}$ is unique. Suppose $e_1$ and $e_2$ are both additive identities. Then $e_1 + e_2 = e_2$ (since $e_1$ is an identity) and $e_1 + e_2 = e_1$ (since $e_2$ is an identity). Therefore $e_1 = e_2$. There is only one additive identity.

Existence-and-uniqueness combined: $\exists!$

The notation $\exists! x, P(x)$ asserts "there exists exactly one $x$ with property $P$." It is equivalent to $(\exists x, P(x)) \wedge (\forall x, \forall y, (P(x) \wedge P(y)) \Rightarrow x = y)$ — existence AND a uniqueness clause. So a full $\exists!$ proof has two parts, like a biconditional proof:

Existence: exhibit (or argue for) a witness. Uniqueness: suppose two satisfiers and show they coincide.

Always do both. Showing existence alone leaves open that ten objects satisfy $P$; showing uniqueness alone leaves open that zero objects do.

Why uniqueness matters — the algebraic perspective

Many algebraic structures are defined up to a uniqueness property. The additive identity is defined to be "the element $e$ with $a + e = a$ for all $a$" — and that definition presupposes uniqueness, or there could be two competing "identities." Uniqueness proofs are what justify naming such an element ("the zero," "the limit," "the GCD") rather than calling it "a zero." Whenever you see a definite article in mathematics — "the inverse," "the minimal polynomial," "the kernel" — there is a uniqueness proof in the background.

Pause and think: Why does the standard uniqueness argument start with "suppose $a$ and $b$ both satisfy $P$" rather than "suppose two distinct $a, b$ both satisfy $P$"? What would go wrong with the second phrasing?

Try it

• Existence drill: prove that there is a positive integer $n$ with $2^n > 1000$. Produce a witness and verify.
• Uniqueness drill: prove that the multiplicative identity in $\mathbb{R}$ (the element $e$ with $a \cdot e = a$ for all $a$) is unique. (Hint: mimic the additive-identity argument with $a = e_2$ and $a = e_1$.)
• Predict the structure: write the two SECTION HEADINGS you would use in a proof that "for every real $a$, there is a unique real $b$ with $a + b = 0$."
• Spot the flaw: 'Proof that there is a unique solution to $x + 5 = 12$: clearly $x = 7$ works.' What is missing?
• Non-constructive challenge: prove there exists a real number that is not rational. Try the constructive route (exhibit $\sqrt{2}$; defer the irrationality proof to a citation) vs the non-constructive route (countability argument).

A trap to watch for

The two halves of an $\exists!$ proof are logically independent and both required. The common error is proving existence and stopping, on the implicit grounds that "I gave a formula, so obviously it is the only one." That reasoning is wrong: a formula gives you A solution; uniqueness must be argued separately. Example: "Prove $\exists! n \in \mathbb{N}, n^2 = 1.$" Existence: take $n = 1$. But uniqueness needs argument — $n = -1$ also squares to 1, so unless you restricted the domain to $\mathbb{N}$ (which excludes negatives) the uniqueness claim would be false. Always state the domain clearly and run the suppose-two-show-one argument. Another trap: in the uniqueness step, do NOT begin with "suppose $a \neq b$ both satisfy $P$ for contradiction"; the cleaner pattern is "suppose $a, b$ both satisfy $P$" (no distinctness assumption) and derive $a = b$. The contradiction phrasing introduces extra structure you do not need.

What you now know

You can prove $\exists x, P(x)$ by exhibiting a constructive witness or by a non-constructive argument that one must exist. You can prove uniqueness by "suppose two, show one." You can combine both into a proof of $\exists! x, P(x)$ with two clearly labelled parts. You understand why uniqueness underwrites the definite article in mathematical naming and where the structure appears in real applications — ODE solvers, cryptography, and the foundations of type theory.

References

• Velleman, D. J. (2019). How to Prove It: A Structured Approach (3rd ed.). Cambridge University Press, §3.6.
• Hammack, R. (2018). Book of Proof (3rd ed.). Open-source textbook, ch. 8.
• Bloch, E. D. (2011). Proofs and Fundamentals. Springer, §3.6 (Existence and uniqueness).
• The Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics, §3.11 (Contractibility).
• Pierce, B. C. et al. (2018). Software Foundations, Vol. 1: Logical Foundations. Online textbook (Coq), ch. 'Logic' (ex_unique).