📂Writing

Frequently Used Symbols and Abbreviations in Mathematics

for all, exist, such that

Example¹

For every $\varepsilon \gt 0$, there is an integer N such that $n \ge N$ implies that $d(p_{n},p)<\varepsilon$.

Every positive real number $\varepsilon$, there exists an integer $N$ such that whenever $n$ is greater than some integer $N$, $d(p_{n},p) \lt \varepsilon$ holds.

$$ \forall \varepsilon \gt 0, \exists N \in \mathbb{N}\quad \text{s.t. } n\ge N \implies d(p_{n},p) \lt \varepsilon $$

Explanation

1. $\forall$

   It means ‘for all’ or ‘for every’, and it is interpreted as ‘for every ~’. Therefore, expressions like ‘for $\forall$’ or ‘$\forall$ all’ are incorrect. The $\LaTeX$ syntax is \forall.

2. $\exists$, $\exists !$, $\nexists$

   It implies ’exist’ and is interpreted as ’there exists’. Adding an exclamation mark means ’there exists uniquely’. The $\LaTeX$ syntax for it is \exists. $\nexists$ stands for ‘does not exist’ and its $\LaTeX$ syntax is \nexists.

3. $\text{s.t.}$

   It’s an abbreviation for ‘such that’ and translated into ‘such’, ’like that’, ‘as follows’. It is not specifically implemented in $\LaTeX$, but if you want to use it, you can type \text{s.t.}.

q.e.d

An abbreviation for the Latin ‘Quod Erat Demonstrandum(QED)’. Literally it translates to ‘what was to be demonstrated’, and figuratively it means ’end of proof’. Often textbooks mark this with a □ or ■. At the fresh shrimp sushi restaurant, ■ is used to denote the end of a section, including proofs.