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 Nov 19 at 15:05
OppoInfinity
1719
add a comment |
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 Nov 19 at 15:05
OppoInfinity
1719
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
Nov 19 at 15:46
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 Nov 19 at 15:05
OppoInfinity
1719
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 Nov 19 at 15:05
OppoInfinity
1719
asked Nov 19 at 15:05
OppoInfinity
1719
asked Nov 19 at 15:05
OppoInfinity
1719
asked Nov 19 at 15:05
OppoInfinity
1719
asked Nov 19 at 15:05
OppoInfinity
1719
1719
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
Nov 19 at 15:46
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
Nov 19 at 15:46
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
Nov 19 at 15:46
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
Nov 19 at 15:46
add a comment |
1 Answer
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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 Nov 19 at 15:48
Ross Millikan
288k23195365
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 Nov 19 at 15:48
Ross Millikan
288k23195365
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 Nov 19 at 15:48
Ross Millikan
288k23195365
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 Nov 19 at 15:48
Ross Millikan
288k23195365
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 Nov 19 at 15:48
Ross Millikan
288k23195365
answered Nov 19 at 15:48
Ross Millikan
288k23195365
answered Nov 19 at 15:48
Ross Millikan
288k23195365
answered Nov 19 at 15:48
Ross Millikan
288k23195365
288k23195365
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
Nov 19 at 15:46