As a lazy mathematician, I'm tired of having to write "for at least one". Similar to the $\forall$ (for all) symbol, does a symbol exist to denote "for at least one"? I've been abbreviating to f.a.l.o., but I was hoping for some more elegant notation.
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$\begingroup$ I have seen many people write "f.s. $x$" to mean "for some $x$." $\endgroup$– Trevor WilsonSep 16, 2012 at 15:47
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$\begingroup$ Yes indeed, it is the following $\exists(x)$ (there exist(s) x). $\endgroup$– amWhyNov 4, 2017 at 19:54
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2 Answers
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The existential quantifier "$\exists x.\phi(x)$" in formal logic denotes "there exists at least one $x$ that satisfies the property $\phi(x)$".
If you're not writing very formally symbolic logic, you should also consider sticking to English, but just writing "some" instead of "at least one", as in
Now, by the Fundamental Theorem of Algebra, $p(z)=0$ for some $z$. ...
since "at least one" is the conventional mathematical meaning of "some"+singular noun. With this wording you're even allowed to keep referring to the $z$ that makes $p(z)=0$ in the following sentences you write, whereas the $z$ in $\exists z.p(z)=0$ goes out of scope at the end of the formula.
Note that whereas one occasionally sees people abbreviate, for example, "$f(x)>0$ for all $x$" as "$f(x)>0 ~\forall x$", this heinous abuse of symbolism is rare for the existential quantifier. Properly used, quantifiers are always written before the formula they control: $$ \forall x.f(x)>0 $$ $$ \exists z.p(z)=0 $$ and so forth. Several minor variations in punctuation exist, such as $$ (\forall x)\, f(x)>0 $$ $$ \forall x(\,f(x)>0\,) $$
answered Sep 16, 2012 at 14:19
hmakholm left over Monicahmakholm left over Monica
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2$\begingroup$ Fantastic answer which not only addressed my question but expanded my knowledge on more general mathematical notation. Thank you very much! I had stumbled upon the $\exists$ symbol, but I wasn't sure whether it implied the existence of a unique object, of multiple objects, or of at least one object. This clears that up. ***** $\endgroup$– lodhbSep 16, 2012 at 15:24
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$\begingroup$ Follow-up question: how could I use the $\exists$ symbol in the following sense: $f(A_0)=\{b\mid b=f(a)$ for at least one $ a \in A_0\}$? $\endgroup$– lodhbSep 16, 2012 at 15:32
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5$\begingroup$ The existence of a unique object is often notated with an exclamation sign combined with $\exists$, as in $\exists_! x.\phi(x)$. Though very useful, it is not taught quite as often as the ordinary $\forall$ and $\exists$ -- and then usually as an abbreviation for asserting existence and uniqueness separately: $$\exists_! x.\phi(x) \text{ means }(\exists x.\phi(x)\land \forall x.\forall y.(\phi(x)\land\phi(y)\to x=y)$$. $\endgroup$ Sep 16, 2012 at 15:32
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1$\begingroup$ @lodhb: Follow-up answer: You don't. If you want to put the specification of $a$ after the property it satisfies, you need to use English (or a natural language of your choice), not symbols. $\endgroup$ Sep 16, 2012 at 15:33
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2$\begingroup$ What’s wrong with $f(A_0) = \{ b \mid \exists a \in A_0.b = f(a) \}$? $\endgroup$ Sep 16, 2012 at 17:39
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Based on what I've read, it sounds like there is no standard abbreviation of "for at least one" or "for some", but what I typically use to denote "for some" in my own personal notes is the inverted F: $\Finv$ (LaTeX \Finv)
It felt like a natural fit since it resembles the existential symbol $\exists$ (whose meaning it shares) just with the top line removed, and "F" is the first sound in the phrase "for some". I commonly use it when I wish to put an existential quantifier after what it quantifies over (improper as it may be), e.g. $$ x+1 = 2 \quad \Finv\ x $$ I considered using $\exists$ for this purpose, but that felt awkward since "for some" has no initial "E" sound, and it would be confusing to have to remember to read $\exists$ as "for some" only when it is placed after a formula.
