8

Is there a symbol for "given" in mathematics? For example, for the statement:

Each member, $x$, of the integer sequence $f(n)$ equals the sum of the two previous members, $f(n-1)$ and $f(n-2)$, given $f(0) = 0$ and $f(1) = 1$.

How do you write this symbolically?

Tyler Durden
  • 954
  • 4
    Actually, I wouldn't use "given" in this context for the natural language version. – Hagen von Eitzen Apr 23 '15 at 14:56
  • Using $\Rightarrow$ for "if.. then..." should be working. – MonkeyKing Apr 23 '15 at 15:00
  • 1
    As a side note, symbols should only be used instead of their natural language counterpart (if they have one) if they improve the readability or clarity of a statement. In this case substituting "given" with a symbol wouldn't do either. – A.P. Apr 23 '15 at 16:01
  • Why does everything have to have a symbol? Maybe you think it makes it easier for those who don't read English. If that was me, I'd much rather see English words than a bunch of strange symbols strung together in a seemingly random way. – Robert Soupe Apr 24 '15 at 02:42
  • But... have you thought of a simple, humble comma? $f(n) = f(n - 1) + f(n - 2)$, $f(0) = 0$, $f(1) = 1$. They do this sort of thing in the OEIS all the time. – Robert Soupe Apr 24 '15 at 02:44
  • What does "given" mean in your sentence? It looks grammatically wrong. Also what is the meaning of $x$ in the middle of the sentence? ", $x$, " – Stef Oct 05 '23 at 21:51

4 Answers4

9

Some common symbols used to express the notion of "given" or "such that" are the colon ":" and the vertical bar "|".

I guess your statement could then be rephrased as:

$x \in \{ f(n),\ n \in \mathbb{N}\ |\ f(n) = f(n-1) + f(n-2),\ f(0) = 0,\ f(1) = 1 \} $

kaslusimoes
  • 223
2

$$\forall n\Big((n\in\Bbb N\wedge n\ge 2)\to f(n)=f(n-1)+f(n-2)\Big) \wedge f(0)=0 \wedge f(1)=1$$

ajotatxe
  • 65,084
2

You can read $$A \rightarrow B$$ as "$A$ implies $B$" and $$A \leftarrow B$$ as "$A$ if $B$" or "$A$ given $B$."

For example, the statement "$n! = 1$ given that $n=0$" can be written $$n! = 1 \leftarrow n=0.$$

goblin GONE
  • 67,744
0

A simple logical and should do the job: $$ f(n) = f(n-1) + f(n-2) \wedge f(0) = 0 \wedge f(1) = 1 $$ These function values act only as additional requirements, they are not required to satisfy $f(n) = f(n-1) + f(n-2)$. Otherwise you could use a $\Rightarrow$.

mvw
  • 34,562