Is there a symbol for "given" in mathematics?
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?
$\endgroup$ 54 Answers
$\begingroup$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:
$\endgroup$ $\begingroup$$x \in \{ f(n),\ n \in \mathbb{N}\ |\ f(n) = f(n-1) + f(n-2),\ f(0) = 0,\ f(1) = 1 \} $
You can read $$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.$$
$\endgroup$ $\begingroup$$$\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$$
$\endgroup$ $\begingroup$A simple logical and should do the job: $$ 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 $\Rightarrow$.
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