What is uniqueness quantification?

Question

Can someone explain the concept of Uniqueness quantification ∃! in an easily understandable way since I can't understand the definition of it, what's special about it with other logical operators like and, or, not?

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Answer 1

I think the "Uniqueness quantification" Wikipedia article does a fair job of answering this question, but try this: how many integers are there which, when added to $3$ give you $7$? Only one, namely $4$. But how many integers are there which, when squared give you $25$? Two! ($5$ and $-5$.)

The uniqueness operator is a mathematical convention which allows us to describe the phenomenon which occurs when there is one and only one mathematical object which satisfies the given conditions.

In practice, you will usually see it paired with the existence operator "$\exists$". This might look like the following: $\forall x \in \mathbb{Z}$, $\exists ! y \in \mathbb{Z}$ such that $x+y=0$. Translation: "For every integer $x$, there exists a unique integer $y$ such that $x+y=0$ (This is the additive inverse property).