A function is a tuple of the form from $X$ to $Y$ $(X,Y,G)$ where $X$ and $Y$ are sets and $G$ is functional of $X\times Y$.

A subset $G$ of $X\times Y$ is called functional if for any $x\in X$ there is a unique $y\in Y$ such that $(x,y)\in G$.

Can you give a example of functional?

  • $\begingroup$ Can you explain where you are having problems with this question? $\endgroup$ – Q the Platypus Nov 10 '16 at 1:15
  • $\begingroup$ I could'nt understand $G$,i.e., definiton of functional. $\endgroup$ – PozcuKushimotoStreet Nov 10 '16 at 1:19
  • $\begingroup$ Why should we know this $G$ in the functions? $\endgroup$ – PozcuKushimotoStreet Nov 10 '16 at 1:21
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    $\begingroup$ Note that some authors don't bother with the tuples, but simply define $G$ itself to be the function. (See e.g. this answer I wrote.) This simplifies notation somewhat, but has the disadvantage (in some people's opinion) that such functions (what your textbook's author apparently calls functional relations) don't explicitly define their codomain ($Y$ in your notation). (The domain $X$ is implicitly defined by $G$, as $X=\{x:(x,y)\in G\}$, but $R=\{y:(x,y)\in G\}$ only defines the range of $G$, which may be a proper subset of the codomain $Y$.) $\endgroup$ – Ilmari Karonen Nov 10 '16 at 14:16

In mathematics a $f$ function acts sort of like a machine where one puts a value in as an argument $f(z)$ and it returns some other value.

The definition of a function tells us what it does. $(X,Y,G)$ In this definition the set $X$ is called the domain and indicates what inputs the function can take.

$Y$ is called the codomain and indicates what outputs the function may return.

$G$ tells us what the function does. It is a set of pairs $(x,y)$ where $x$ is an element of $X$ and $y$ is an element of Y. When an argument $z$ is inputted into the function the function will find the $(x,y)$ pair where $x = z$ and return the $y$ element from that pair.

In order for $G$ to work correctly there needs to an $(x,y)$ pair for every element in $X$ and there can't be two pairs with the same $x$ value because then the function doesn't know which one to use.

A very simple example of a function would be if you had the the following sets.

$$X = \{0,1\}$$ $$Y = \{2,3\}$$ $$G = \{(0,2),(1,3)\}$$ $$f = (X,Y,G) $$

In this $f(0) = 2$ and $f(1) = 3$. The definition of functional is a set that has the properties stable for being used like this. A set of pairs with a unique domain member of the pair for each element of the domain.


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