Followup on Proof: Equivalence Classes
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The original question at hand was:
Suppose α and β are equivalence relations on the set S. Suppose further that the relation γ is defined as follows: For x,y∈S, xγy means xαy and xβy. Prove that γ is also an equivalence relation on S.
I'm not quite sure where I should start on this, but I think it has something to do with both α and β being equivalence relations, and then proving that γ is also an equivalence relation through the use of reflexive, symmetric, and transitive properties. Any ideas?
(already answered in another thread)
The followup to this question is: Suppose, in the preceding problem, S = {1, 2, 3, 4, 5, 6}, the equivalence classes of α are {1, 2, 3} and {4, 5, 6} and the equivalence classes of β are {1, 4}, {2, 5}, and {3, 6}. What are the equivalence classes of γ?
Any ideas on this one? I have no idea how to approach this.
discrete-mathematics equivalence-relations
asked Nov 17 at 18:47
NKP
31
add a comment |
up vote
0
down vote
favorite
The original question at hand was:
Suppose α and β are equivalence relations on the set S. Suppose further that the relation γ is defined as follows: For x,y∈S, xγy means xαy and xβy. Prove that γ is also an equivalence relation on S.
I'm not quite sure where I should start on this, but I think it has something to do with both α and β being equivalence relations, and then proving that γ is also an equivalence relation through the use of reflexive, symmetric, and transitive properties. Any ideas?
(already answered in another thread)
The followup to this question is: Suppose, in the preceding problem, S = {1, 2, 3, 4, 5, 6}, the equivalence classes of α are {1, 2, 3} and {4, 5, 6} and the equivalence classes of β are {1, 4}, {2, 5}, and {3, 6}. What are the equivalence classes of γ?
Any ideas on this one? I have no idea how to approach this.
discrete-mathematics equivalence-relations
asked Nov 17 at 18:47
NKP
31
Just go through the axioms one by one. For example we have $xgamma x$ if and only if $x alpha x$ and $xbeta x$. But we know...
– leibnewtz
Nov 17 at 18:53
Could you give an example of this?
– NKP
Nov 17 at 19:30
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The original question at hand was:
Suppose α and β are equivalence relations on the set S. Suppose further that the relation γ is defined as follows: For x,y∈S, xγy means xαy and xβy. Prove that γ is also an equivalence relation on S.
I'm not quite sure where I should start on this, but I think it has something to do with both α and β being equivalence relations, and then proving that γ is also an equivalence relation through the use of reflexive, symmetric, and transitive properties. Any ideas?
(already answered in another thread)
The followup to this question is: Suppose, in the preceding problem, S = {1, 2, 3, 4, 5, 6}, the equivalence classes of α are {1, 2, 3} and {4, 5, 6} and the equivalence classes of β are {1, 4}, {2, 5}, and {3, 6}. What are the equivalence classes of γ?
Any ideas on this one? I have no idea how to approach this.
discrete-mathematics equivalence-relations
asked Nov 17 at 18:47
NKP
31
The original question at hand was:
Suppose α and β are equivalence relations on the set S. Suppose further that the relation γ is defined as follows: For x,y∈S, xγy means xαy and xβy. Prove that γ is also an equivalence relation on S.
I'm not quite sure where I should start on this, but I think it has something to do with both α and β being equivalence relations, and then proving that γ is also an equivalence relation through the use of reflexive, symmetric, and transitive properties. Any ideas?
(already answered in another thread)
The followup to this question is: Suppose, in the preceding problem, S = {1, 2, 3, 4, 5, 6}, the equivalence classes of α are {1, 2, 3} and {4, 5, 6} and the equivalence classes of β are {1, 4}, {2, 5}, and {3, 6}. What are the equivalence classes of γ?
Any ideas on this one? I have no idea how to approach this.
discrete-mathematics equivalence-relations
discrete-mathematics equivalence-relations
asked Nov 17 at 18:47
NKP
31
asked Nov 17 at 18:47
NKP
31
asked Nov 17 at 18:47
NKP
31
asked Nov 17 at 18:47
NKP
31
asked Nov 17 at 18:47
NKP
31
31
Just go through the axioms one by one. For example we have $xgamma x$ if and only if $x alpha x$ and $xbeta x$. But we know...
– leibnewtz
Nov 17 at 18:53
Could you give an example of this?
– NKP
Nov 17 at 19:30
add a comment |
Just go through the axioms one by one. For example we have $xgamma x$ if and only if $x alpha x$ and $xbeta x$. But we know...
– leibnewtz
Nov 17 at 18:53
Could you give an example of this?
– NKP
Nov 17 at 19:30
Just go through the axioms one by one. For example we have $xgamma x$ if and only if $x alpha x$ and $xbeta x$. But we know...
– leibnewtz
Nov 17 at 18:53
Just go through the axioms one by one. For example we have $xgamma x$ if and only if $x alpha x$ and $xbeta x$. But we know...
– leibnewtz
Nov 17 at 18:53
Could you give an example of this?
– NKP
Nov 17 at 19:30
Could you give an example of this?
– NKP
Nov 17 at 19:30
add a comment |
1 Answer
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For the first question, check the axioms one by one. For example (i) reflexive: Take any $xin S$. $xalpha x$ and $xbeta x$, thus... (continue)
For the second part, start with one element in $S$, e.g. with $1$ and then add all elements which are equivelant to $1$. If you have all of them, continue and do the same thing again. This is a valid strategy, since $S$ is finite. Make sure that you understand why an equivalence relation partitions its set into classes.
Btw, you will learn more from your problem sheets, if you don't ask for solutions online. So be strong next time.
answered Nov 17 at 20:00
N.Beck
1665
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
For the first question, check the axioms one by one. For example (i) reflexive: Take any $xin S$. $xalpha x$ and $xbeta x$, thus... (continue)
For the second part, start with one element in $S$, e.g. with $1$ and then add all elements which are equivelant to $1$. If you have all of them, continue and do the same thing again. This is a valid strategy, since $S$ is finite. Make sure that you understand why an equivalence relation partitions its set into classes.
Btw, you will learn more from your problem sheets, if you don't ask for solutions online. So be strong next time.
answered Nov 17 at 20:00
N.Beck
1665
add a comment |
up vote
0
down vote
For the first question, check the axioms one by one. For example (i) reflexive: Take any $xin S$. $xalpha x$ and $xbeta x$, thus... (continue)
For the second part, start with one element in $S$, e.g. with $1$ and then add all elements which are equivelant to $1$. If you have all of them, continue and do the same thing again. This is a valid strategy, since $S$ is finite. Make sure that you understand why an equivalence relation partitions its set into classes.
Btw, you will learn more from your problem sheets, if you don't ask for solutions online. So be strong next time.
answered Nov 17 at 20:00
N.Beck
1665
add a comment |
up vote
0
down vote
up vote
0
down vote
For the first question, check the axioms one by one. For example (i) reflexive: Take any $xin S$. $xalpha x$ and $xbeta x$, thus... (continue)
For the second part, start with one element in $S$, e.g. with $1$ and then add all elements which are equivelant to $1$. If you have all of them, continue and do the same thing again. This is a valid strategy, since $S$ is finite. Make sure that you understand why an equivalence relation partitions its set into classes.
Btw, you will learn more from your problem sheets, if you don't ask for solutions online. So be strong next time.
answered Nov 17 at 20:00
N.Beck
1665
For the first question, check the axioms one by one. For example (i) reflexive: Take any $xin S$. $xalpha x$ and $xbeta x$, thus... (continue)
For the second part, start with one element in $S$, e.g. with $1$ and then add all elements which are equivelant to $1$. If you have all of them, continue and do the same thing again. This is a valid strategy, since $S$ is finite. Make sure that you understand why an equivalence relation partitions its set into classes.
Btw, you will learn more from your problem sheets, if you don't ask for solutions online. So be strong next time.
answered Nov 17 at 20:00
N.Beck
1665
answered Nov 17 at 20:00
N.Beck
1665
answered Nov 17 at 20:00
N.Beck
1665
answered Nov 17 at 20:00
N.Beck
1665
1665
add a comment |
add a comment |
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Just go through the axioms one by one. For example we have $xgamma x$ if and only if $x alpha x$ and $xbeta x$. But we know...
– leibnewtz
Nov 17 at 18:53
Could you give an example of this?
– NKP
Nov 17 at 19:30