Surjective function at a subset of its codomain
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In Wikipedia we read:
In mathematics, a function $f$ from a set $X$ to a set $Y$ is surjective (or onto), or a surjection, if for every element $y$ in the codomain $Y$ of $f$ there is at least one element $x$ in the domain $X$ of $f$ such that $f(x) = y$.
Can a function $f:Ato B$ be called surjective at a specific subset of its codomain?
I ask if this definition exists:
A function $f$ from a set $X$ to a set $Y$ is surjective at a subset $A$ of its codomain if for every element $y$ in the set $A$ there is at least one element $x$ in the domain $X$ of $f$ such that $f(x) = y$.
For example the function $f:mathbb{R}tomathbb{R}$
such that $f(x)=x^2$ is not surjective (at its codomain $mathbb{R}$).
But is it called surjective at $mathbb{R}_+$ for example?(or each subset of $mathbb{R}_+$)
By that definition every function is surjective at all subsets of its image.
functions
asked Jan 11 at 20:11
JohnJohn
514
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add a comment |
$begingroup$
In Wikipedia we read:
In mathematics, a function $f$ from a set $X$ to a set $Y$ is surjective (or onto), or a surjection, if for every element $y$ in the codomain $Y$ of $f$ there is at least one element $x$ in the domain $X$ of $f$ such that $f(x) = y$.
Can a function $f:Ato B$ be called surjective at a specific subset of its codomain?
I ask if this definition exists:
A function $f$ from a set $X$ to a set $Y$ is surjective at a subset $A$ of its codomain if for every element $y$ in the set $A$ there is at least one element $x$ in the domain $X$ of $f$ such that $f(x) = y$.
For example the function $f:mathbb{R}tomathbb{R}$
such that $f(x)=x^2$ is not surjective (at its codomain $mathbb{R}$).
But is it called surjective at $mathbb{R}_+$ for example?(or each subset of $mathbb{R}_+$)
By that definition every function is surjective at all subsets of its image.
functions
asked Jan 11 at 20:11
JohnJohn
514
$endgroup$
1
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"By that definition every function is surjective at all subsets of its image." Not quite: only at its image. It's not a well-defined function for any smaller subset.
$endgroup$
– user3482749
Jan 11 at 20:16
1
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We just say that $A$ is a subset of the image of $f$. I never heard about this definition.
$endgroup$
– Yanko
Jan 11 at 20:21
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No, absolutely unusual: A function is surjective at its image or a subset of it.
$endgroup$
– Wuestenfux
Jan 11 at 20:26
add a comment |
$begingroup$
In Wikipedia we read:
In mathematics, a function $f$ from a set $X$ to a set $Y$ is surjective (or onto), or a surjection, if for every element $y$ in the codomain $Y$ of $f$ there is at least one element $x$ in the domain $X$ of $f$ such that $f(x) = y$.
Can a function $f:Ato B$ be called surjective at a specific subset of its codomain?
I ask if this definition exists:
A function $f$ from a set $X$ to a set $Y$ is surjective at a subset $A$ of its codomain if for every element $y$ in the set $A$ there is at least one element $x$ in the domain $X$ of $f$ such that $f(x) = y$.
For example the function $f:mathbb{R}tomathbb{R}$
such that $f(x)=x^2$ is not surjective (at its codomain $mathbb{R}$).
But is it called surjective at $mathbb{R}_+$ for example?(or each subset of $mathbb{R}_+$)
By that definition every function is surjective at all subsets of its image.
functions
asked Jan 11 at 20:11
JohnJohn
514
$endgroup$
In Wikipedia we read:
In mathematics, a function $f$ from a set $X$ to a set $Y$ is surjective (or onto), or a surjection, if for every element $y$ in the codomain $Y$ of $f$ there is at least one element $x$ in the domain $X$ of $f$ such that $f(x) = y$.
Can a function $f:Ato B$ be called surjective at a specific subset of its codomain?
I ask if this definition exists:
A function $f$ from a set $X$ to a set $Y$ is surjective at a subset $A$ of its codomain if for every element $y$ in the set $A$ there is at least one element $x$ in the domain $X$ of $f$ such that $f(x) = y$.
For example the function $f:mathbb{R}tomathbb{R}$
such that $f(x)=x^2$ is not surjective (at its codomain $mathbb{R}$).
But is it called surjective at $mathbb{R}_+$ for example?(or each subset of $mathbb{R}_+$)
By that definition every function is surjective at all subsets of its image.
functions
functions
asked Jan 11 at 20:11
JohnJohn
514
asked Jan 11 at 20:11
JohnJohn
514
asked Jan 11 at 20:11
JohnJohn
514
asked Jan 11 at 20:11
JohnJohn
514
asked Jan 11 at 20:11
JohnJohn
514
514
1
$begingroup$
"By that definition every function is surjective at all subsets of its image." Not quite: only at its image. It's not a well-defined function for any smaller subset.
$endgroup$
– user3482749
Jan 11 at 20:16
1
$begingroup$
We just say that $A$ is a subset of the image of $f$. I never heard about this definition.
$endgroup$
– Yanko
Jan 11 at 20:21
$begingroup$
No, absolutely unusual: A function is surjective at its image or a subset of it.
$endgroup$
– Wuestenfux
Jan 11 at 20:26
add a comment |
1
$begingroup$
"By that definition every function is surjective at all subsets of its image." Not quite: only at its image. It's not a well-defined function for any smaller subset.
$endgroup$
– user3482749
Jan 11 at 20:16
1
$begingroup$
We just say that $A$ is a subset of the image of $f$. I never heard about this definition.
$endgroup$
– Yanko
Jan 11 at 20:21
$begingroup$
No, absolutely unusual: A function is surjective at its image or a subset of it.
$endgroup$
– Wuestenfux
Jan 11 at 20:26
1
1
$begingroup$
"By that definition every function is surjective at all subsets of its image." Not quite: only at its image. It's not a well-defined function for any smaller subset.
$endgroup$
– user3482749
Jan 11 at 20:16
$begingroup$
"By that definition every function is surjective at all subsets of its image." Not quite: only at its image. It's not a well-defined function for any smaller subset.
$endgroup$
– user3482749
Jan 11 at 20:16
1
1
$begingroup$
We just say that $A$ is a subset of the image of $f$. I never heard about this definition.
$endgroup$
– Yanko
Jan 11 at 20:21
$begingroup$
We just say that $A$ is a subset of the image of $f$. I never heard about this definition.
$endgroup$
– Yanko
Jan 11 at 20:21
$begingroup$
No, absolutely unusual: A function is surjective at its image or a subset of it.
$endgroup$
– Wuestenfux
Jan 11 at 20:26
$begingroup$
No, absolutely unusual: A function is surjective at its image or a subset of it.
$endgroup$
– Wuestenfux
Jan 11 at 20:26
add a comment |
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1
$begingroup$
"By that definition every function is surjective at all subsets of its image." Not quite: only at its image. It's not a well-defined function for any smaller subset.
$endgroup$
– user3482749
Jan 11 at 20:16
1
$begingroup$
We just say that $A$ is a subset of the image of $f$. I never heard about this definition.
$endgroup$
– Yanko
Jan 11 at 20:21
$begingroup$
No, absolutely unusual: A function is surjective at its image or a subset of it.
$endgroup$
– Wuestenfux
Jan 11 at 20:26