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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
asked yesterday
Unknown MathMan
828
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up vote
1
down 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
asked yesterday
Unknown MathMan
828
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
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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
asked yesterday
Unknown MathMan
828
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
asked yesterday
Unknown MathMan
828
asked yesterday
Unknown MathMan
828
asked yesterday
Unknown MathMan
828
asked yesterday
Unknown MathMan
828
asked yesterday
Unknown MathMan
828
828
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
add a comment |
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
add a comment |
1 Answer
1
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oldest
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1
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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.
answered yesterday
Ross Millikan
287k23195364
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
answered yesterday
Ross Millikan
287k23195364
add a comment |
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.
answered yesterday
Ross Millikan
287k23195364
add a comment |
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.
answered yesterday
Ross Millikan
287k23195364
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.
answered yesterday
Ross Millikan
287k23195364
answered yesterday
Ross Millikan
287k23195364
answered yesterday
Ross Millikan
287k23195364
answered yesterday
Ross Millikan
287k23195364
287k23195364
add a comment |
add a comment |
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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