My text tells me that the general term of a sequence can be looked at like a function:

$ f:\mathbb{N}\rightarrow \mathbb{R} $

What does that mean translated into common english?


This specific case is called a sequence. The function $f$ takes in an input from the natural numbers (denoted $\mathbb N$), and gives an output in the real numbers (denoted $\mathbb R$).

In general, this notation contains three parts: the function name, the domain, and the codomain. The function name (in your case $f$) is the same $f$ appearing in the notation $f(x)$. The domain is the set of inputs to your function (in your case the natural numbers $\mathbb N$). The codomain is the set of possible outputs that the function can give (in your case the real number $\mathbb R$).

  • $\begingroup$ In high school I learned about the range of the function. How is that related to the codomain? $\endgroup$ – User1996 Sep 7 '14 at 18:30
  • $\begingroup$ The range will typically be a subset of the codomain. The elements of the range are the values that are mapped to. For instance, you could define $f:\mathbb R \to \mathbb R$, $f(x) = 12$. The codomain of your function is $\mathbb R$, but the range is just $\{12\}$. $\endgroup$ – theage Sep 7 '14 at 18:32
  • $\begingroup$ Ah I see, so the codomain is the family of numbers that the range belongs to? $\endgroup$ – User1996 Sep 7 '14 at 18:35
  • $\begingroup$ Exactly!${}{}{}$ $\endgroup$ – theage Sep 7 '14 at 18:37

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