In the preceding post (real number system 5), we assumed any point on a real number line was corresponding to a real number with no fail. Nobody will not believe it. If two sets (in the case, a real number line and real numbers) have some kind of relationship, there will be a function. Let us define a function.
[function, domain, and range]
A relation from a set $X$ to a set $Y$ which is many to one or one to one is called a function from $X$ to $Y$. Then, a set $X$ is said to be the domain of the function and a set $Y$ is said to be the range of the function. These relations are written by
\[ f : X\rightarrow Y \]
or, in familiar form,
\[ f(x)=y,\quad (x\in X, y\in Y) \]
Here, $x$ is said to be a argument of the function, and $y$ is said to be a value of the function.
The expressions, which you have known enough, could be extended in general. For instance, the following forms are available.
\[ f : X\times X\rightarrow Y, \quad or\quad f(x_1,x_2)=y,\quad (x_1,x_2\in X, y\in Y) \]
If every element of the range of the function is related to some elements of the domain of the function, the function is surjective, or is said to be the mapping from $X$ onto $Y$. That is, for all $y\in Y$, there are some $x$ such that $f(x)=y$. Do not confuse, it is not always true that $x$ is one. The term of 'mapping ' is equivalent to 'function '.
If one element of the range of the function is related to only one element of the domain of the function, the function is one to one, or injective, or is said to be the mapping from $X$ into $Y$. That is, if $f(x_1)=f(x_2)$, then $x_1=x_2$. The contrapositive of the statement is that if $x_1\ne x_2$, then $f(x_1)\ne f(x_2)$.
Especially, two sets $X$ and $Y$ are said to be in one-to-one correspondence if there exists a one to one surjective function with domain $X$ and range $Y$. A One-to-one correspondence means the mapping from $X$ onto and into $Y$. Hence, two sets in one-to-one correspondence have the same number of elements.
[function, domain, and range]
A relation from a set $X$ to a set $Y$ which is many to one or one to one is called a function from $X$ to $Y$. Then, a set $X$ is said to be the domain of the function and a set $Y$ is said to be the range of the function. These relations are written by
\[ f : X\rightarrow Y \]
or, in familiar form,
\[ f(x)=y,\quad (x\in X, y\in Y) \]
Here, $x$ is said to be a argument of the function, and $y$ is said to be a value of the function.
The expressions, which you have known enough, could be extended in general. For instance, the following forms are available.
\[ f : X\times X\rightarrow Y, \quad or\quad f(x_1,x_2)=y,\quad (x_1,x_2\in X, y\in Y) \]
If every element of the range of the function is related to some elements of the domain of the function, the function is surjective, or is said to be the mapping from $X$ onto $Y$. That is, for all $y\in Y$, there are some $x$ such that $f(x)=y$. Do not confuse, it is not always true that $x$ is one. The term of 'mapping ' is equivalent to 'function '.
If one element of the range of the function is related to only one element of the domain of the function, the function is one to one, or injective, or is said to be the mapping from $X$ into $Y$. That is, if $f(x_1)=f(x_2)$, then $x_1=x_2$. The contrapositive of the statement is that if $x_1\ne x_2$, then $f(x_1)\ne f(x_2)$.
Especially, two sets $X$ and $Y$ are said to be in one-to-one correspondence if there exists a one to one surjective function with domain $X$ and range $Y$. A One-to-one correspondence means the mapping from $X$ onto and into $Y$. Hence, two sets in one-to-one correspondence have the same number of elements.
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