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I encountered the following statement in my linear algebra textbook:

Let $S$ be any nonempty set and $F$ be any field, and let $G(S,F)$ denote the set of all functions from $S$ to $F$.

I understand what fields, sets, and functions are. However, I do not understand the language used in this statement. Specifically, I do not understand what is meant by, "let $G(S,F)$ denote the set of all functions from $S$ to $F$". The closest similarity of such language that I've heard is when discussing (open and closed) intervals.

I would greatly appreciate it if someone could please take the time to clarify the meaning of this statement.

Thank you.

2 Answers 2

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EDIT: The set of all functions from $S$ to $F$ means the set of all functions with domain $S$ and codomain $F$.

A function from $S$ to $F$ can be viewed as a subset of $S\times F$. Specifically, a function from $S$ to $F$ is a subset $R$ of $S\times F$ such that for every $s\in S$ there exists a unique $f\in F$ such that $(s,f)\in R$. The set of all functions from $S$ to $F$ is the set of all such subsets of $G(S,F)$. The fact that $F$ is a field is not relevant for this 'construction'.

Put differently, if you know what a function from $S$ to $F$ is and what a set is, then the set of all functions from $S$ to $F$ can be constructed from the powerset of $S\times F$ by the axiom of separation.

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    Is your usage of $S\times F$ denoting the cross product? – 2017-01-05
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    The usual product of sets. What would a cross product of sets be? – 2017-01-05
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    Unfortunately, I haven't taken any set theory. This is just what I encountered in my introductory linear algebra textbook. I only know what a set itself is at the most basic level. – 2017-01-05
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    Then my answer doesn't quite answer your question, I guess. What part of *"Let $G(S,F)$ denote the set of all functions from $S$ to $F$."* is not clear? Is it clear what a function from $S$ to $F$ is? – 2017-01-05
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    Yes, I think that's my misunderstanding. How can we have a range of functions? That doesn't make sense to me. – 2017-01-05
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    On a second reading, I think I might understand where the problem lies. A function *from* $S$ *to* $F$ means a function with *domain* $S$ and *codomain* $F$. – 2017-01-05
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    Ok, I think that was the problem. I was interpreting it as some kind of range, rather than a mapping. Thank you for your assistance. :) – 2017-01-05
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Let $S = \{A, B\}$.
Let $F = \{0, 1\}$.

There is one function from $S$ to $F$, and let's call this function $g_1$, that has these properties:

$$g_1 (A) = 0$$ $$g_1 (B) = 0$$

There is another function from $S$ to $F$, which we'll call $g_2$, that has the following properties:

$$g_2 (A) = 0$$ $$g_2 (B) = 1$$

Believe it or not, there are just two other distinct functions from $S$ to $F$. I won't list them, but believe me when I say that $G = \{g_1 , g_2 . g_3 . g_4\}$ is the set of all functions from $S$ to $F$. The book is using a notation that is much more practical than listing all of the functions, as that number can get unmanageable quickly!