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What is $\exists !$ equivalent to? I need to write $\exists !x \,P(x)$ without using the $\exists !$ symbol; thus, I am wondering what the $\exists !$ symbol is equivalent to.

Daniel W. Farlow
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Torched90
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2 Answers2

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We say that there exists a unique $x$ with property $P$ provided two things are true. First, there exists $x$ with property $P$, and second, for all $x_1$ and $x_2$, if $x_1$ and $x_2$ both have property $P$, then $x_1$ and $x_2$ are the same. In other words, the logical statement $(\exists !x)(P(x))$ is defined by $$ [(\exists x)(P(x))]\land[(\forall x_1)(\forall x_2)(P(x_1)\land P(x_2))\to (x_1=x_2)].\tag{1} $$ This is what the unique existential quantifier, $\exists !$, is logically equivalent to.

Example: For all $a\in\mathbb{R}$, there exists a unique $b\in\mathbb{R}$ such that $a+b=0$. [Reword this statement of unique existence in a form like that communicated in $(1)$.]

Meaning. For all $a\in\mathbb{R}$, there exists $b\in\mathbb{R}$ such that $a+b=0$, and for all $b_1,b_2\in\mathbb{R}$, if $a+b_1=0$ and $a+b_2=0$, then $b_1=b_2$.

Daniel W. Farlow
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  • I would encourage you to think about that in terms of language first and then quantifiers. Regardless, in general, you have $$\neg(\exists x)(P(x)) \equiv (\forall x)(\neg P(x)).$$ Similarly, you also have $$\neg(\forall x)(P(x)) \equiv (\exists x)(\neg P(x)).$$ So the negation, $\neg$, sort of "moves through" the expression. I actually asked a generalization of this kind of question not long ago on MSE: http://math.stackexchange.com/questions/1095530/negation-of-quantifiers – Daniel W. Farlow Jan 26 '15 at 03:08
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$$\exists!x\,P(x)\Leftrightarrow\exists x\forall y[P(y)\leftrightarrow y=x]$$

bof
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