Exercise 563Logic, sets and proofs › Quantifiers and negationL1Added 3 Nov 2026

The uniqueness quantifier

#uniqueness#exists unique#quantifier

Statement

Write "there exists a unique real $x$ such that $x^2=0$ " in formal notation, and prove it.

Hint

Use $\exists!$ for "there exists a unique".

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Notation. $\exists! x\in\mathbb{R},\ x^2=0$ .

Proof.
- *Existence.* $x=0$ satisfies $x^2=0$ .
- *Uniqueness.* If $x^2=0$ then $x=0$ .

Conclusion. The statement holds.

Remark. $\exists!x,\ P(x)$ is equivalent to $\exists x,\ P(x)\land\forall y,\ (P(y)\Rightarrow y=x)$ .