Function: Difference between revisions
Created page with "A '''function''' is a many-to-one map between two sets. == Formal definition == Let $D,R \neq \emptyset$ be sets. If $$ \forall x \in D: \exists y \in R, f(x)=y $$ Then $f$ is a function of '''domain''' $D$ and '''range''' $R$, noted as $f: D \to R$. '''Example''': $$ f(x)=2x^2, x \in \mathbb{C} $$ is a function, where $f: \mathbb{C} \to \mathbb{C}$" |
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A '''function''' is a many-to-one map between two [[set|sets]]. | A '''function''' is a many(one)-to-one map between two [[set|sets]]. | ||
== Formal definition == | == Formal definition == | ||
Let $ | Let $D_f,C_f,R_f \neq \emptyset$ be [[set|sets]]. | ||
If | If | ||
$$ | $$ | ||
\forall x \in | \forall x \in D_f: \exists y \in C_f, f(x)=y, | ||
$$ | $$ | ||
then $f$ is a function of '''domain''' $D_f$ and '''codomain''' $C_f$, noted as $f: D_f \to C_f$; | |||
Furthermore, | |||
if | |||
$$ | |||
\forall y \in R_f \subseteq C_f, \exists x \in D_f, f(x)=y, | |||
$$ | |||
then $R_f$ is the '''range''' of $f$; | |||
Lastly, | |||
if | |||
$$ | |||
\forall y \in R_f: \exists f^{-1}, f^{-1}(y)=x, | |||
$$ | |||
then function $f$ is said to be '''reversible''' and the '''bijection''' of $D$ to $R_f$. $f$ and $f^{-1}$ are said to be '''inverse functions''' of each other. | |||
<blockquote> | |||
While a function can be many-to-one, a bijection is an one-to-one function of its domain to its range. | |||
</blockquote> | |||
'''Example''': | '''Example''': | ||
$$ | $$ | ||
f(x)=2x^2, x \in \mathbb{ | f(x)=2x^2, x \in \mathbb{R} | ||
$$ | |||
is a function, where $f: \mathbb{R} \to \mathbb{R}$. | |||
It is however not a bijection from $\mathbb{R}$ to $\mathbb{R}$, as $\exists y \in \mathbb{R}, \nexists f^{-1}, f^{-1}(y)=x$, such as $y=-1$ | |||
<blockquote> | |||
Note the range $R_f \subset \mathbb{R}$, as $\exists y \in \mathbb{R}: y \notin R_f$, such as $y=-1$ | |||
</blockquote> | |||
<ref>{{#cite:Q2199}}</ref> | |||
== Image and preimage == | |||
Given function $f: A \to B$ | |||
If for $A_1 \subseteq A, B_1 \subseteq B$, | |||
$$ | |||
B_1=f(A_1)=\{f(a)|\forall a \in A_1\} | |||
$$ | $$ | ||
is | Then $B_1$ is known as the '''image''' of $A_1$ in $B$ under $f$, and $A_1$ is known as the '''pre-image''' of $B_1$ under $f$, noted $f^{\leftarrow}(B_1)=A_1$ | ||
<ref>{{#cite:Q2199}}</ref> | |||
Latest revision as of 12:07, 16 October 2026
A function is a many(one)-to-one map between two sets.
Formal definition
Let $D_f,C_f,R_f \neq \emptyset$ be sets.
If
$$ \forall x \in D_f: \exists y \in C_f, f(x)=y, $$
then $f$ is a function of domain $D_f$ and codomain $C_f$, noted as $f: D_f \to C_f$;
Furthermore,
if
$$ \forall y \in R_f \subseteq C_f, \exists x \in D_f, f(x)=y, $$
then $R_f$ is the range of $f$;
Lastly,
if
$$ \forall y \in R_f: \exists f^{-1}, f^{-1}(y)=x, $$
then function $f$ is said to be reversible and the bijection of $D$ to $R_f$. $f$ and $f^{-1}$ are said to be inverse functions of each other.
While a function can be many-to-one, a bijection is an one-to-one function of its domain to its range.
Example:
$$ f(x)=2x^2, x \in \mathbb{R} $$
is a function, where $f: \mathbb{R} \to \mathbb{R}$.
It is however not a bijection from $\mathbb{R}$ to $\mathbb{R}$, as $\exists y \in \mathbb{R}, \nexists f^{-1}, f^{-1}(y)=x$, such as $y=-1$
Note the range $R_f \subset \mathbb{R}$, as $\exists y \in \mathbb{R}: y \notin R_f$, such as $y=-1$
Image and preimage
Given function $f: A \to B$
If for $A_1 \subseteq A, B_1 \subseteq B$,
$$ B_1=f(A_1)=\{f(a)|\forall a \in A_1\} $$
Then $B_1$ is known as the image of $A_1$ in $B$ under $f$, and $A_1$ is known as the pre-image of $B_1$ under $f$, noted $f^{\leftarrow}(B_1)=A_1$