Reflexive, symmetric, and transitive closures
$begingroup$
The problem: 
I am having difficulty with this problem. How do I even start? I know what reflexive, symmetric and transitive closures intuitively mean but I am struggling to find s(r(R)), (symmetric closure OF reflexive closure of R) for example. Do I even have to find out all the specific elements of s(r(R)) or do I just use a more general method?
I deduced that all elements of R are as follows: elements in R, where the columns are primes and the rows are the nth multiple of the prime. (Can someone verify this?) 
And I know that r(R) (reflexive closure of R) is the union of R with the set { (0,0), (1,1) , (2,2), (3,3) ... }.
But I am having difficulty listing out all the elements in s(r(R)) and r(s(R)) so that I can solve part (a), for example.
Any help would be much appreciated. Thanks.
relations order-theory
edited Dec 6 '18 at 12:27
Graham Kemp
86.4k43479
asked Dec 6 '18 at 7:34
R. RudinthkinR. Rudinthkin
1
$endgroup$
add a comment |
$begingroup$
The problem:
I am having difficulty with this problem. How do I even start? I know what reflexive, symmetric and transitive closures intuitively mean but I am struggling to find s(r(R)), (symmetric closure OF reflexive closure of R) for example. Do I even have to find out all the specific elements of s(r(R)) or do I just use a more general method?
I deduced that all elements of R are as follows: elements in R, where the columns are primes and the rows are the nth multiple of the prime. (Can someone verify this?)
And I know that r(R) (reflexive closure of R) is the union of R with the set { (0,0), (1,1) , (2,2), (3,3) ... }.
But I am having difficulty listing out all the elements in s(r(R)) and r(s(R)) so that I can solve part (a), for example.
Any help would be much appreciated. Thanks.
relations order-theory
edited Dec 6 '18 at 12:27
Graham Kemp
86.4k43479
asked Dec 6 '18 at 7:34
R. RudinthkinR. Rudinthkin
1
$endgroup$
add a comment |
$begingroup$
The problem:
I am having difficulty with this problem. How do I even start? I know what reflexive, symmetric and transitive closures intuitively mean but I am struggling to find s(r(R)), (symmetric closure OF reflexive closure of R) for example. Do I even have to find out all the specific elements of s(r(R)) or do I just use a more general method?
I deduced that all elements of R are as follows: elements in R, where the columns are primes and the rows are the nth multiple of the prime. (Can someone verify this?)
And I know that r(R) (reflexive closure of R) is the union of R with the set { (0,0), (1,1) , (2,2), (3,3) ... }.
But I am having difficulty listing out all the elements in s(r(R)) and r(s(R)) so that I can solve part (a), for example.
Any help would be much appreciated. Thanks.
relations order-theory
edited Dec 6 '18 at 12:27
Graham Kemp
86.4k43479
asked Dec 6 '18 at 7:34
R. RudinthkinR. Rudinthkin
1
$endgroup$
The problem:
I am having difficulty with this problem. How do I even start? I know what reflexive, symmetric and transitive closures intuitively mean but I am struggling to find s(r(R)), (symmetric closure OF reflexive closure of R) for example. Do I even have to find out all the specific elements of s(r(R)) or do I just use a more general method?
I deduced that all elements of R are as follows: elements in R, where the columns are primes and the rows are the nth multiple of the prime. (Can someone verify this?)
And I know that r(R) (reflexive closure of R) is the union of R with the set { (0,0), (1,1) , (2,2), (3,3) ... }.
But I am having difficulty listing out all the elements in s(r(R)) and r(s(R)) so that I can solve part (a), for example.
Any help would be much appreciated. Thanks.
relations order-theory
relations order-theory
edited Dec 6 '18 at 12:27
Graham Kemp
86.4k43479
asked Dec 6 '18 at 7:34
R. RudinthkinR. Rudinthkin
1
edited Dec 6 '18 at 12:27
Graham Kemp
86.4k43479
asked Dec 6 '18 at 7:34
R. RudinthkinR. Rudinthkin
1
edited Dec 6 '18 at 12:27
Graham Kemp
86.4k43479
edited Dec 6 '18 at 12:27
Graham Kemp
86.4k43479
edited Dec 6 '18 at 12:27
Graham Kemp
86.4k43479
86.4k43479
asked Dec 6 '18 at 7:34
R. RudinthkinR. Rudinthkin
1
asked Dec 6 '18 at 7:34
R. RudinthkinR. Rudinthkin
1
asked Dec 6 '18 at 7:34
R. RudinthkinR. Rudinthkin
1
1
add a comment |
add a comment |
1 Answer
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$begingroup$
$R$ is the set of paired natural numbers $(x,y)$ where $y=px$ for some $p$ which is a prime number.$$R={(x,px):xinBbb N, pinBbb P}$$
And I know that r(R) (reflexive closure of R) is the union of R with the set { (0,0), (1,1) , (2,2), (3,3) ... }.
Yes. Thus the reflexive closure is therefore defined:$$begin{split}r(R):&={(x,px):xinBbb N, pinBbb P}cup{(x,x):xinBbb N}\&= {(x,qx):xinBbb N,qinBbb Pcup{1}}end{split}$$... or in words: the set of paired natural numbers $(x,y)$ where $y=qx$ for some $q$ which is a prime number or $1$.
In the same manner, describe the symmetric closure, $s(R)$, and likewise $r(s(R))$ and $s(r(R))$.
$endgroup$
add a comment |
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$begingroup$
$R$ is the set of paired natural numbers $(x,y)$ where $y=px$ for some $p$ which is a prime number.$$R={(x,px):xinBbb N, pinBbb P}$$
And I know that r(R) (reflexive closure of R) is the union of R with the set { (0,0), (1,1) , (2,2), (3,3) ... }.
Yes. Thus the reflexive closure is therefore defined:$$begin{split}r(R):&={(x,px):xinBbb N, pinBbb P}cup{(x,x):xinBbb N}\&= {(x,qx):xinBbb N,qinBbb Pcup{1}}end{split}$$... or in words: the set of paired natural numbers $(x,y)$ where $y=qx$ for some $q$ which is a prime number or $1$.
In the same manner, describe the symmetric closure, $s(R)$, and likewise $r(s(R))$ and $s(r(R))$.
$endgroup$
add a comment |
$begingroup$
$R$ is the set of paired natural numbers $(x,y)$ where $y=px$ for some $p$ which is a prime number.$$R={(x,px):xinBbb N, pinBbb P}$$
And I know that r(R) (reflexive closure of R) is the union of R with the set { (0,0), (1,1) , (2,2), (3,3) ... }.
Yes. Thus the reflexive closure is therefore defined:$$begin{split}r(R):&={(x,px):xinBbb N, pinBbb P}cup{(x,x):xinBbb N}\&= {(x,qx):xinBbb N,qinBbb Pcup{1}}end{split}$$... or in words: the set of paired natural numbers $(x,y)$ where $y=qx$ for some $q$ which is a prime number or $1$.
In the same manner, describe the symmetric closure, $s(R)$, and likewise $r(s(R))$ and $s(r(R))$.
$endgroup$
add a comment |
$begingroup$
$R$ is the set of paired natural numbers $(x,y)$ where $y=px$ for some $p$ which is a prime number.$$R={(x,px):xinBbb N, pinBbb P}$$
And I know that r(R) (reflexive closure of R) is the union of R with the set { (0,0), (1,1) , (2,2), (3,3) ... }.
Yes. Thus the reflexive closure is therefore defined:$$begin{split}r(R):&={(x,px):xinBbb N, pinBbb P}cup{(x,x):xinBbb N}\&= {(x,qx):xinBbb N,qinBbb Pcup{1}}end{split}$$... or in words: the set of paired natural numbers $(x,y)$ where $y=qx$ for some $q$ which is a prime number or $1$.
In the same manner, describe the symmetric closure, $s(R)$, and likewise $r(s(R))$ and $s(r(R))$.
$endgroup$
$R$ is the set of paired natural numbers $(x,y)$ where $y=px$ for some $p$ which is a prime number.$$R={(x,px):xinBbb N, pinBbb P}$$
And I know that r(R) (reflexive closure of R) is the union of R with the set { (0,0), (1,1) , (2,2), (3,3) ... }.
Yes. Thus the reflexive closure is therefore defined:$$begin{split}r(R):&={(x,px):xinBbb N, pinBbb P}cup{(x,x):xinBbb N}\&= {(x,qx):xinBbb N,qinBbb Pcup{1}}end{split}$$... or in words: the set of paired natural numbers $(x,y)$ where $y=qx$ for some $q$ which is a prime number or $1$.
In the same manner, describe the symmetric closure, $s(R)$, and likewise $r(s(R))$ and $s(r(R))$.
answered Dec 6 '18 at 13:02
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Graham Kemp
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