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Equation in question:

\[ \forall n \in \mathbb{N} : n^2 \ge n \]

The symbol $:$ is a symbol for "such that"
$\forall$ for all, for any, for each

When do you know when to use: for all, for any, for each? Does it matter which one you use, or are there rules to follow when selecting one?

I say it's something like:

$n^2 \ge n$ is true for all/for any/for each $n \in \mathbb{N}$

But, the $:$ means such that, so then it changes everything, for me. Because then it could be

$n^2 \ge n$ is true for all such that $n \in \mathbb{N}$? I'm so confused.

k170
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Jimmy Jhoners
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    Style point: One probably should write, in this such that context, $n\le n_2$. And if the colon is being used for such that (not universal) then some of the sentence is missing, so best English reading is not clear. As it stands it means nothing. – André Nicolas Jul 16 '14 at 02:04
  • I've never seen : used to mean "such that". If I were reading aloud, I'd say something like "For all n in the natural numbers, n(squared?) is greater than or equal to n". – hardmath Jul 16 '14 at 02:04
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    @hardmath a colon is often used in "set builder notation" e.g. ${n\in\mathbb{N} : n>3}$ which can be read "For all natural numbers n such that n>3". – mathematician Jul 16 '14 at 02:20
  • I ask because according to wikipedia, ∀ is used for all, for any or for each, and the example was: ∀ x: (P)x where (P)x is true for all of x, but the colon is there, but there was no "such that" – Jimmy Jhoners Jul 16 '14 at 02:23
  • The colon is also used as a separator by some people, instead of parentheses. – André Nicolas Jul 16 '14 at 02:53
  • How are you supposed to differentiate between them? – Jimmy Jhoners Jul 16 '14 at 02:56
  • Personally, I would write this as "for each $n \in {\mathbb N},$ we have $n \leq n^2$" or some variation of this. In most cases I see no reason to excessively clutter the statement with symbols such as $\forall$ unless the focus is on quantification issues or the Borel classification of a set. – Dave L. Renfro Jul 16 '14 at 18:43

2 Answers2

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There are two basic logical "quantifiers" commonly in use. These symbols help specify when a statement is true. Let's take a look at your original statement.

$$∀n∈ℕ:n^2≥n$$

This says "For all natural numbers, $n^2\ge n$". It is not specifying that n is a natural number greater than or equal to $n^2$, it is making a logical statement

$$P(n) = n^2\ge n$$

and then asserts that $P(n)$ is true for each natural number n. In general,

$$\forall x\in S: Q(x) $$

is a logical statement that says $Q(x)$ holds for all x in S.

The other logical quantifier is $\exists$, and if $P(x)$ is a logical statement, then

$$\exists x\in S: P(x) $$

is read "There exists some x in S such that P(x)", and it means that somewhere within S there is an x that fulfills P(x).

Clay Thomas
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The general construction imo is

$$ \begin{array}{ccccc} \forall & A & : & B\\ \hline \textrm{For all} & A & \textrm{follows that} & B & \textrm{is true}\\ \textrm{For all} & A & \textrm{the statement} & B & \textrm{is true} \end{array} $$

johannesvalks
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