Why equality of relations is defined like that?











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Suppose $R_1$ is a $n$-ary relation which is a subset of $prod_{i=1}^n A_i$ and $R_2$ is a $m$-ary relation which is a subset of $prod_{i=1}^m B_i$. The equality for two relations is defined as, $R_1=R_2$ if $n=m$ and $A_i=B_i$ for all $1le ile n$.




So, if we consider $A={1,2}$ and $B={a,b}$ and $C={a,b,c}$. Now, suppose $R_1$ is the relation, which is a subset of $Atimes B$ is ${(1,a),(2,b)}$ and $R_2$ is a subset of $Atimes C$ and also ${(1,a),(2,b)}$. Here as $Bneq C$, we don't have $R_1=R_2$ from the definition! But as a set they are same!



Can anyone explain me why equality of relation is defined like that? In other word if we have defined $R_1=R_2$ if they are equal as set then where we are doing wrong? am I missing something about definition of relation?






discrete-mathematics relations







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  • You have missed a statement in your definition that the ordered pairs in the two relations must be the same. It isn't important to your question.
    – Ross Millikan
    yesterday















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Suppose $R_1$ is a $n$-ary relation which is a subset of $prod_{i=1}^n A_i$ and $R_2$ is a $m$-ary relation which is a subset of $prod_{i=1}^m B_i$. The equality for two relations is defined as, $R_1=R_2$ if $n=m$ and $A_i=B_i$ for all $1le ile n$.




So, if we consider $A={1,2}$ and $B={a,b}$ and $C={a,b,c}$. Now, suppose $R_1$ is the relation, which is a subset of $Atimes B$ is ${(1,a),(2,b)}$ and $R_2$ is a subset of $Atimes C$ and also ${(1,a),(2,b)}$. Here as $Bneq C$, we don't have $R_1=R_2$ from the definition! But as a set they are same!



Can anyone explain me why equality of relation is defined like that? In other word if we have defined $R_1=R_2$ if they are equal as set then where we are doing wrong? am I missing something about definition of relation?






discrete-mathematics relations







share|cite|improve this question






















  • You have missed a statement in your definition that the ordered pairs in the two relations must be the same. It isn't important to your question.
    – Ross Millikan
    yesterday













up vote
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Suppose $R_1$ is a $n$-ary relation which is a subset of $prod_{i=1}^n A_i$ and $R_2$ is a $m$-ary relation which is a subset of $prod_{i=1}^m B_i$. The equality for two relations is defined as, $R_1=R_2$ if $n=m$ and $A_i=B_i$ for all $1le ile n$.




So, if we consider $A={1,2}$ and $B={a,b}$ and $C={a,b,c}$. Now, suppose $R_1$ is the relation, which is a subset of $Atimes B$ is ${(1,a),(2,b)}$ and $R_2$ is a subset of $Atimes C$ and also ${(1,a),(2,b)}$. Here as $Bneq C$, we don't have $R_1=R_2$ from the definition! But as a set they are same!



Can anyone explain me why equality of relation is defined like that? In other word if we have defined $R_1=R_2$ if they are equal as set then where we are doing wrong? am I missing something about definition of relation?






discrete-mathematics relations







share|cite|improve this question














Suppose $R_1$ is a $n$-ary relation which is a subset of $prod_{i=1}^n A_i$ and $R_2$ is a $m$-ary relation which is a subset of $prod_{i=1}^m B_i$. The equality for two relations is defined as, $R_1=R_2$ if $n=m$ and $A_i=B_i$ for all $1le ile n$.




So, if we consider $A={1,2}$ and $B={a,b}$ and $C={a,b,c}$. Now, suppose $R_1$ is the relation, which is a subset of $Atimes B$ is ${(1,a),(2,b)}$ and $R_2$ is a subset of $Atimes C$ and also ${(1,a),(2,b)}$. Here as $Bneq C$, we don't have $R_1=R_2$ from the definition! But as a set they are same!



Can anyone explain me why equality of relation is defined like that? In other word if we have defined $R_1=R_2$ if they are equal as set then where we are doing wrong? am I missing something about definition of relation?






discrete-mathematics relations




discrete-mathematics relations






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Unknown MathMan

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  • You have missed a statement in your definition that the ordered pairs in the two relations must be the same. It isn't important to your question.
    – Ross Millikan
    yesterday


















  • You have missed a statement in your definition that the ordered pairs in the two relations must be the same. It isn't important to your question.
    – Ross Millikan
    yesterday
















You have missed a statement in your definition that the ordered pairs in the two relations must be the same. It isn't important to your question.
– Ross Millikan
yesterday




You have missed a statement in your definition that the ordered pairs in the two relations must be the same. It isn't important to your question.
– Ross Millikan
yesterday










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Your are correct that as sets $R_1$ and $R_2$ are the same. As you say, our definition of equality in relations demands that the domain and range sets be the same. A relation is defined as a triple of domain, range, and the pairs in the relation. In your example $R_1$ is surjective and $R_2$ is not.






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    Your are correct that as sets $R_1$ and $R_2$ are the same. As you say, our definition of equality in relations demands that the domain and range sets be the same. A relation is defined as a triple of domain, range, and the pairs in the relation. In your example $R_1$ is surjective and $R_2$ is not.






    share|cite|improve this answer

























      up vote
      1
      down vote



      accepted










      Your are correct that as sets $R_1$ and $R_2$ are the same. As you say, our definition of equality in relations demands that the domain and range sets be the same. A relation is defined as a triple of domain, range, and the pairs in the relation. In your example $R_1$ is surjective and $R_2$ is not.






      share|cite|improve this answer























        up vote
        1
        down vote



        accepted







        up vote
        1
        down vote



        accepted






        Your are correct that as sets $R_1$ and $R_2$ are the same. As you say, our definition of equality in relations demands that the domain and range sets be the same. A relation is defined as a triple of domain, range, and the pairs in the relation. In your example $R_1$ is surjective and $R_2$ is not.






        share|cite|improve this answer












        Your are correct that as sets $R_1$ and $R_2$ are the same. As you say, our definition of equality in relations demands that the domain and range sets be the same. A relation is defined as a triple of domain, range, and the pairs in the relation. In your example $R_1$ is surjective and $R_2$ is not.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered yesterday









        Ross Millikan

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