Set theory, functions and inverses











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I'm doing an intro course on set theory and have the question if the inverses of the surjective functions in the sets A=｛a, b｝and B= {c, d, e} are also functions.



So far, I thought that the inverse of a function f(x) for example, is f⁻¹(y), meaning that every function also has an inverse (which is also a function).
Given that this would make the question rather redundant, I'm not quite sure in my assumption anymore, so I would be glad, if someone could verify or falsify (and explain it properly) it.






functions elementary-set-theory inverse-function







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  • Inverse defined from the range of the given function is a function.
    – Thomas Shelby
    yesterday















up vote
0
down vote

favorite












I'm doing an intro course on set theory and have the question if the inverses of the surjective functions in the sets A=｛a, b｝and B= {c, d, e} are also functions.



So far, I thought that the inverse of a function f(x) for example, is f⁻¹(y), meaning that every function also has an inverse (which is also a function).
Given that this would make the question rather redundant, I'm not quite sure in my assumption anymore, so I would be glad, if someone could verify or falsify (and explain it properly) it.






functions elementary-set-theory inverse-function







share|cite|improve this question







New contributor




K. Meyer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • Inverse defined from the range of the given function is a function.
    – Thomas Shelby
    yesterday













up vote
0
down vote

favorite









up vote
0
down vote

favorite











I'm doing an intro course on set theory and have the question if the inverses of the surjective functions in the sets A=｛a, b｝and B= {c, d, e} are also functions.



So far, I thought that the inverse of a function f(x) for example, is f⁻¹(y), meaning that every function also has an inverse (which is also a function).
Given that this would make the question rather redundant, I'm not quite sure in my assumption anymore, so I would be glad, if someone could verify or falsify (and explain it properly) it.






functions elementary-set-theory inverse-function







share|cite|improve this question







New contributor




K. Meyer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











I'm doing an intro course on set theory and have the question if the inverses of the surjective functions in the sets A=｛a, b｝and B= {c, d, e} are also functions.



So far, I thought that the inverse of a function f(x) for example, is f⁻¹(y), meaning that every function also has an inverse (which is also a function).
Given that this would make the question rather redundant, I'm not quite sure in my assumption anymore, so I would be glad, if someone could verify or falsify (and explain it properly) it.






functions elementary-set-theory inverse-function




functions elementary-set-theory inverse-function






share|cite|improve this question







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K. Meyer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question







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K. Meyer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









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  • Inverse defined from the range of the given function is a function.
    – Thomas Shelby
    yesterday


















  • Inverse defined from the range of the given function is a function.
    – Thomas Shelby
    yesterday
















Inverse defined from the range of the given function is a function.
– Thomas Shelby
yesterday




Inverse defined from the range of the given function is a function.
– Thomas Shelby
yesterday










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A relation is a set of ordered pairs, and it's inverse is the following relation:
$$R^{-1}:={(y,x)|(x,y)in R}$$
And $R$ is called a function if $forall x,y_1,y_2$, $(x,y_1)in R land (x,y_2)in R implies y_1=y_2$. In the case of functions, we can always define a formal inverse, and we call a function $f$ injective if it's formal inverse is also a function. And not all of the surjections are injections, for example let
$$f:={(a,c),(b,d),(a,e)}$$
Now, $f$ is a surjection from $A$ to $B$, but it's not an injection, because it's inverse is not a function.






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    A relation is a set of ordered pairs, and it's inverse is the following relation:
    $$R^{-1}:={(y,x)|(x,y)in R}$$
    And $R$ is called a function if $forall x,y_1,y_2$, $(x,y_1)in R land (x,y_2)in R implies y_1=y_2$. In the case of functions, we can always define a formal inverse, and we call a function $f$ injective if it's formal inverse is also a function. And not all of the surjections are injections, for example let
    $$f:={(a,c),(b,d),(a,e)}$$
    Now, $f$ is a surjection from $A$ to $B$, but it's not an injection, because it's inverse is not a function.






    share|cite|improve this answer

























      up vote
      0
      down vote













      A relation is a set of ordered pairs, and it's inverse is the following relation:
      $$R^{-1}:={(y,x)|(x,y)in R}$$
      And $R$ is called a function if $forall x,y_1,y_2$, $(x,y_1)in R land (x,y_2)in R implies y_1=y_2$. In the case of functions, we can always define a formal inverse, and we call a function $f$ injective if it's formal inverse is also a function. And not all of the surjections are injections, for example let
      $$f:={(a,c),(b,d),(a,e)}$$
      Now, $f$ is a surjection from $A$ to $B$, but it's not an injection, because it's inverse is not a function.






      share|cite|improve this answer























        up vote
        0
        down vote










        up vote
        0
        down vote









        A relation is a set of ordered pairs, and it's inverse is the following relation:
        $$R^{-1}:={(y,x)|(x,y)in R}$$
        And $R$ is called a function if $forall x,y_1,y_2$, $(x,y_1)in R land (x,y_2)in R implies y_1=y_2$. In the case of functions, we can always define a formal inverse, and we call a function $f$ injective if it's formal inverse is also a function. And not all of the surjections are injections, for example let
        $$f:={(a,c),(b,d),(a,e)}$$
        Now, $f$ is a surjection from $A$ to $B$, but it's not an injection, because it's inverse is not a function.






        share|cite|improve this answer












        A relation is a set of ordered pairs, and it's inverse is the following relation:
        $$R^{-1}:={(y,x)|(x,y)in R}$$
        And $R$ is called a function if $forall x,y_1,y_2$, $(x,y_1)in R land (x,y_2)in R implies y_1=y_2$. In the case of functions, we can always define a formal inverse, and we call a function $f$ injective if it's formal inverse is also a function. And not all of the surjections are injections, for example let
        $$f:={(a,c),(b,d),(a,e)}$$
        Now, $f$ is a surjection from $A$ to $B$, but it's not an injection, because it's inverse is not a function.







        share|cite|improve this answer












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