Are all functions mappings between sets?
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I am trying to better understand what a function is.
I often see diagrams with two ovals with a bunch of lines between them, and I believe these represent sets, but I don't know if there's an official name for this, nor if this is how all functions behave.
Would it be accurate to say that a set is just a collection of objects without duplicates, and that a function is a mapping between "input" objects of one set and "output" objects of another set, such that each input is related to exactly one output object?
algebra-precalculus functions elementary-set-theory definition
edited Aug 30 at 1:35
Andrés E. Caicedo
63.4k7154238
asked Aug 30 at 1:08
user525966
1,544619
 |Â
show 5 more comments
up vote
0
down vote
favorite
I am trying to better understand what a function is.
I often see diagrams with two ovals with a bunch of lines between them, and I believe these represent sets, but I don't know if there's an official name for this, nor if this is how all functions behave.
Would it be accurate to say that a set is just a collection of objects without duplicates, and that a function is a mapping between "input" objects of one set and "output" objects of another set, such that each input is related to exactly one output object?
algebra-precalculus functions elementary-set-theory definition
edited Aug 30 at 1:35
Andrés E. Caicedo
63.4k7154238
asked Aug 30 at 1:08
user525966
1,544619
Yes, that's right.
– Michael Biro
Aug 30 at 1:11
1
Yes, that is exactly right. More formally, we might treat a function as a subset of the cartesian product $Xtimes Y$ where $X$ is our "domain" (set of inputs) and $Y$ is our "codomain" (set of possible outputs) such that every element in the domain appears exactly once as the first entry in an ordered pair. In that case we actually have a function $f$ being of the form something like $(x_1,y_1),(x_2,y_2),(x_3,y_3),dots$. For shorthand we usually rewrite $(x_1,y_1)in f$ as instead $f(x_1)=y_1$.
– JMoravitz
Aug 30 at 1:12
Note: no such requirement is placed on the elements from the codomain. Those functions which do happen to have every element in the codomain appearing at least once each, or at most once each are given a special name: surjective and injective functions respectively or bijective if both.
– JMoravitz
Aug 30 at 1:15
@JMoravitz So the "cartesian product" is also a set, since we're writing things like $(x_1, y_1) in f$? How would this correspond to writing the function as something like $f(x) = x^2$ or something? Is this another way of writing it or a transformation on the set notation in some form?
– user525966
Aug 30 at 1:18
1
A minor frustration when it comes to being rigorous, not only is the rule for how inputs are changed into outputs important for a function, but the domain and codomain are also important for adequately describing the function. Note for example $f~:~Bbb ZtoBbb Z$ s.t. $f(x)=x^2$ is neither injective nor surjective. Meanwhile $f~:~Bbb Nto Bbb Z$ s.t. $f(x)=x^2$ is injective but not surjective. Finally $f~:~Bbb R^+toBbb R^+$ s.t. $f(x)=x^2$ is both injective and surjective.
– JMoravitz
Aug 30 at 1:25
 |Â
show 5 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to better understand what a function is.
I often see diagrams with two ovals with a bunch of lines between them, and I believe these represent sets, but I don't know if there's an official name for this, nor if this is how all functions behave.
Would it be accurate to say that a set is just a collection of objects without duplicates, and that a function is a mapping between "input" objects of one set and "output" objects of another set, such that each input is related to exactly one output object?
algebra-precalculus functions elementary-set-theory definition
edited Aug 30 at 1:35
Andrés E. Caicedo
63.4k7154238
asked Aug 30 at 1:08
user525966
1,544619
I am trying to better understand what a function is.
I often see diagrams with two ovals with a bunch of lines between them, and I believe these represent sets, but I don't know if there's an official name for this, nor if this is how all functions behave.
Would it be accurate to say that a set is just a collection of objects without duplicates, and that a function is a mapping between "input" objects of one set and "output" objects of another set, such that each input is related to exactly one output object?
algebra-precalculus functions elementary-set-theory definition
algebra-precalculus functions elementary-set-theory definition
edited Aug 30 at 1:35
Andrés E. Caicedo
63.4k7154238
asked Aug 30 at 1:08
user525966
1,544619
edited Aug 30 at 1:35
Andrés E. Caicedo
63.4k7154238
asked Aug 30 at 1:08
user525966
1,544619
edited Aug 30 at 1:35
Andrés E. Caicedo
63.4k7154238
edited Aug 30 at 1:35
Andrés E. Caicedo
63.4k7154238
edited Aug 30 at 1:35
Andrés E. Caicedo
63.4k7154238
63.4k7154238
asked Aug 30 at 1:08
user525966
1,544619
asked Aug 30 at 1:08
user525966
1,544619
asked Aug 30 at 1:08
user525966
1,544619
1,544619
Yes, that's right.
– Michael Biro
Aug 30 at 1:11
1
Yes, that is exactly right. More formally, we might treat a function as a subset of the cartesian product $Xtimes Y$ where $X$ is our "domain" (set of inputs) and $Y$ is our "codomain" (set of possible outputs) such that every element in the domain appears exactly once as the first entry in an ordered pair. In that case we actually have a function $f$ being of the form something like $(x_1,y_1),(x_2,y_2),(x_3,y_3),dots$. For shorthand we usually rewrite $(x_1,y_1)in f$ as instead $f(x_1)=y_1$.
– JMoravitz
Aug 30 at 1:12
Note: no such requirement is placed on the elements from the codomain. Those functions which do happen to have every element in the codomain appearing at least once each, or at most once each are given a special name: surjective and injective functions respectively or bijective if both.
– JMoravitz
Aug 30 at 1:15
@JMoravitz So the "cartesian product" is also a set, since we're writing things like $(x_1, y_1) in f$? How would this correspond to writing the function as something like $f(x) = x^2$ or something? Is this another way of writing it or a transformation on the set notation in some form?
– user525966
Aug 30 at 1:18
1
A minor frustration when it comes to being rigorous, not only is the rule for how inputs are changed into outputs important for a function, but the domain and codomain are also important for adequately describing the function. Note for example $f~:~Bbb ZtoBbb Z$ s.t. $f(x)=x^2$ is neither injective nor surjective. Meanwhile $f~:~Bbb Nto Bbb Z$ s.t. $f(x)=x^2$ is injective but not surjective. Finally $f~:~Bbb R^+toBbb R^+$ s.t. $f(x)=x^2$ is both injective and surjective.
– JMoravitz
Aug 30 at 1:25
 |Â
show 5 more comments
Yes, that's right.
– Michael Biro
Aug 30 at 1:11
1
Yes, that is exactly right. More formally, we might treat a function as a subset of the cartesian product $Xtimes Y$ where $X$ is our "domain" (set of inputs) and $Y$ is our "codomain" (set of possible outputs) such that every element in the domain appears exactly once as the first entry in an ordered pair. In that case we actually have a function $f$ being of the form something like $(x_1,y_1),(x_2,y_2),(x_3,y_3),dots$. For shorthand we usually rewrite $(x_1,y_1)in f$ as instead $f(x_1)=y_1$.
– JMoravitz
Aug 30 at 1:12
Note: no such requirement is placed on the elements from the codomain. Those functions which do happen to have every element in the codomain appearing at least once each, or at most once each are given a special name: surjective and injective functions respectively or bijective if both.
– JMoravitz
Aug 30 at 1:15
@JMoravitz So the "cartesian product" is also a set, since we're writing things like $(x_1, y_1) in f$? How would this correspond to writing the function as something like $f(x) = x^2$ or something? Is this another way of writing it or a transformation on the set notation in some form?
– user525966
Aug 30 at 1:18
1
A minor frustration when it comes to being rigorous, not only is the rule for how inputs are changed into outputs important for a function, but the domain and codomain are also important for adequately describing the function. Note for example $f~:~Bbb ZtoBbb Z$ s.t. $f(x)=x^2$ is neither injective nor surjective. Meanwhile $f~:~Bbb Nto Bbb Z$ s.t. $f(x)=x^2$ is injective but not surjective. Finally $f~:~Bbb R^+toBbb R^+$ s.t. $f(x)=x^2$ is both injective and surjective.
– JMoravitz
Aug 30 at 1:25
Yes, that's right.
– Michael Biro
Aug 30 at 1:11
Yes, that's right.
– Michael Biro
Aug 30 at 1:11
1
1
Yes, that is exactly right. More formally, we might treat a function as a subset of the cartesian product $Xtimes Y$ where $X$ is our "domain" (set of inputs) and $Y$ is our "codomain" (set of possible outputs) such that every element in the domain appears exactly once as the first entry in an ordered pair. In that case we actually have a function $f$ being of the form something like $(x_1,y_1),(x_2,y_2),(x_3,y_3),dots$. For shorthand we usually rewrite $(x_1,y_1)in f$ as instead $f(x_1)=y_1$.
– JMoravitz
Aug 30 at 1:12
Yes, that is exactly right. More formally, we might treat a function as a subset of the cartesian product $Xtimes Y$ where $X$ is our "domain" (set of inputs) and $Y$ is our "codomain" (set of possible outputs) such that every element in the domain appears exactly once as the first entry in an ordered pair. In that case we actually have a function $f$ being of the form something like $(x_1,y_1),(x_2,y_2),(x_3,y_3),dots$. For shorthand we usually rewrite $(x_1,y_1)in f$ as instead $f(x_1)=y_1$.
– JMoravitz
Aug 30 at 1:12
Note: no such requirement is placed on the elements from the codomain. Those functions which do happen to have every element in the codomain appearing at least once each, or at most once each are given a special name: surjective and injective functions respectively or bijective if both.
– JMoravitz
Aug 30 at 1:15
Note: no such requirement is placed on the elements from the codomain. Those functions which do happen to have every element in the codomain appearing at least once each, or at most once each are given a special name: surjective and injective functions respectively or bijective if both.
– JMoravitz
Aug 30 at 1:15
@JMoravitz So the "cartesian product" is also a set, since we're writing things like $(x_1, y_1) in f$? How would this correspond to writing the function as something like $f(x) = x^2$ or something? Is this another way of writing it or a transformation on the set notation in some form?
– user525966
Aug 30 at 1:18
@JMoravitz So the "cartesian product" is also a set, since we're writing things like $(x_1, y_1) in f$? How would this correspond to writing the function as something like $f(x) = x^2$ or something? Is this another way of writing it or a transformation on the set notation in some form?
– user525966
Aug 30 at 1:18
1
1
A minor frustration when it comes to being rigorous, not only is the rule for how inputs are changed into outputs important for a function, but the domain and codomain are also important for adequately describing the function. Note for example $f~:~Bbb ZtoBbb Z$ s.t. $f(x)=x^2$ is neither injective nor surjective. Meanwhile $f~:~Bbb Nto Bbb Z$ s.t. $f(x)=x^2$ is injective but not surjective. Finally $f~:~Bbb R^+toBbb R^+$ s.t. $f(x)=x^2$ is both injective and surjective.
– JMoravitz
Aug 30 at 1:25
A minor frustration when it comes to being rigorous, not only is the rule for how inputs are changed into outputs important for a function, but the domain and codomain are also important for adequately describing the function. Note for example $f~:~Bbb ZtoBbb Z$ s.t. $f(x)=x^2$ is neither injective nor surjective. Meanwhile $f~:~Bbb Nto Bbb Z$ s.t. $f(x)=x^2$ is injective but not surjective. Finally $f~:~Bbb R^+toBbb R^+$ s.t. $f(x)=x^2$ is both injective and surjective.
– JMoravitz
Aug 30 at 1:25
 |Â
show 5 more comments
1 Answer
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Your explanation of functions is clear and correct.
A function has three ingredients: Domain, Range,and Assignment.
Domain is the set of inputs, Range is the set of outputs and assignment is a formula or table which tells you what goes to what.
And as you mentioned we do not want to send the same input to two different outputs.
We can have graphs,tables, definitions, or any other way to represent functions.
Examples of real world functions are social security numbers assigned to people, birthdays, age, and so fort.
answered Aug 30 at 2:58
Mohammad Riazi-Kermani
31k41853
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
Your explanation of functions is clear and correct.
A function has three ingredients: Domain, Range,and Assignment.
Domain is the set of inputs, Range is the set of outputs and assignment is a formula or table which tells you what goes to what.
And as you mentioned we do not want to send the same input to two different outputs.
We can have graphs,tables, definitions, or any other way to represent functions.
Examples of real world functions are social security numbers assigned to people, birthdays, age, and so fort.
answered Aug 30 at 2:58
Mohammad Riazi-Kermani
31k41853
add a comment |Â
up vote
0
down vote
accepted
Your explanation of functions is clear and correct.
A function has three ingredients: Domain, Range,and Assignment.
Domain is the set of inputs, Range is the set of outputs and assignment is a formula or table which tells you what goes to what.
And as you mentioned we do not want to send the same input to two different outputs.
We can have graphs,tables, definitions, or any other way to represent functions.
Examples of real world functions are social security numbers assigned to people, birthdays, age, and so fort.
answered Aug 30 at 2:58
Mohammad Riazi-Kermani
31k41853
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
Your explanation of functions is clear and correct.
A function has three ingredients: Domain, Range,and Assignment.
Domain is the set of inputs, Range is the set of outputs and assignment is a formula or table which tells you what goes to what.
And as you mentioned we do not want to send the same input to two different outputs.
We can have graphs,tables, definitions, or any other way to represent functions.
Examples of real world functions are social security numbers assigned to people, birthdays, age, and so fort.
answered Aug 30 at 2:58
Mohammad Riazi-Kermani
31k41853
Your explanation of functions is clear and correct.
A function has three ingredients: Domain, Range,and Assignment.
Domain is the set of inputs, Range is the set of outputs and assignment is a formula or table which tells you what goes to what.
And as you mentioned we do not want to send the same input to two different outputs.
We can have graphs,tables, definitions, or any other way to represent functions.
Examples of real world functions are social security numbers assigned to people, birthdays, age, and so fort.
answered Aug 30 at 2:58
Mohammad Riazi-Kermani
31k41853
answered Aug 30 at 2:58
Mohammad Riazi-Kermani
31k41853
answered Aug 30 at 2:58
Mohammad Riazi-Kermani
31k41853
answered Aug 30 at 2:58
Mohammad Riazi-Kermani
31k41853
31k41853
add a comment |Â
add a comment |Â
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Yes, that's right.
– Michael Biro
Aug 30 at 1:11
1
Yes, that is exactly right. More formally, we might treat a function as a subset of the cartesian product $Xtimes Y$ where $X$ is our "domain" (set of inputs) and $Y$ is our "codomain" (set of possible outputs) such that every element in the domain appears exactly once as the first entry in an ordered pair. In that case we actually have a function $f$ being of the form something like $(x_1,y_1),(x_2,y_2),(x_3,y_3),dots$. For shorthand we usually rewrite $(x_1,y_1)in f$ as instead $f(x_1)=y_1$.
– JMoravitz
Aug 30 at 1:12
Note: no such requirement is placed on the elements from the codomain. Those functions which do happen to have every element in the codomain appearing at least once each, or at most once each are given a special name: surjective and injective functions respectively or bijective if both.
– JMoravitz
Aug 30 at 1:15
@JMoravitz So the "cartesian product" is also a set, since we're writing things like $(x_1, y_1) in f$? How would this correspond to writing the function as something like $f(x) = x^2$ or something? Is this another way of writing it or a transformation on the set notation in some form?
– user525966
Aug 30 at 1:18
1
A minor frustration when it comes to being rigorous, not only is the rule for how inputs are changed into outputs important for a function, but the domain and codomain are also important for adequately describing the function. Note for example $f~:~Bbb ZtoBbb Z$ s.t. $f(x)=x^2$ is neither injective nor surjective. Meanwhile $f~:~Bbb Nto Bbb Z$ s.t. $f(x)=x^2$ is injective but not surjective. Finally $f~:~Bbb R^+toBbb R^+$ s.t. $f(x)=x^2$ is both injective and surjective.
– JMoravitz
Aug 30 at 1:25