Mixing
$ R$ is left-total if and only if $ operatornameId_A subseteq R circ R^-1 $
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Proof:
$$R;text is left-total Longleftrightarrow operatornameId_A subseteq R circ R^-1 $$
$R subseteq A times A $
$ R circ R^-1 = (a,c) mid exists b in A: aRb; text and ; bR^-1c $
This is the third part of this exercise. The first were similar, where you had to prove something if and only if R is transitive. I managed to prove it but I'm struggling with the assignment above quite a bit.
Also could not find anything online, but because I'm German and my assignment is in German, I might not have been able to translate it correctly to find what I need.
discrete-mathematics proof-writing relations
edited Jul 26 at 16:00
Bernard
110k635102
asked Jul 26 at 15:09
Awesome36
82
add a comment |Â
up vote
1
down vote
favorite
Proof:
$$R;text is left-total Longleftrightarrow operatornameId_A subseteq R circ R^-1 $$
$R subseteq A times A $
$ R circ R^-1 = (a,c) mid exists b in A: aRb; text and ; bR^-1c $
This is the third part of this exercise. The first were similar, where you had to prove something if and only if R is transitive. I managed to prove it but I'm struggling with the assignment above quite a bit.
Also could not find anything online, but because I'm German and my assignment is in German, I might not have been able to translate it correctly to find what I need.
discrete-mathematics proof-writing relations
edited Jul 26 at 16:00
Bernard
110k635102
asked Jul 26 at 15:09
Awesome36
82
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Proof:
$$R;text is left-total Longleftrightarrow operatornameId_A subseteq R circ R^-1 $$
$R subseteq A times A $
$ R circ R^-1 = (a,c) mid exists b in A: aRb; text and ; bR^-1c $
This is the third part of this exercise. The first were similar, where you had to prove something if and only if R is transitive. I managed to prove it but I'm struggling with the assignment above quite a bit.
Also could not find anything online, but because I'm German and my assignment is in German, I might not have been able to translate it correctly to find what I need.
discrete-mathematics proof-writing relations
edited Jul 26 at 16:00
Bernard
110k635102
asked Jul 26 at 15:09
Awesome36
82
Proof:
$$R;text is left-total Longleftrightarrow operatornameId_A subseteq R circ R^-1 $$
$R subseteq A times A $
$ R circ R^-1 = (a,c) mid exists b in A: aRb; text and ; bR^-1c $
This is the third part of this exercise. The first were similar, where you had to prove something if and only if R is transitive. I managed to prove it but I'm struggling with the assignment above quite a bit.
Also could not find anything online, but because I'm German and my assignment is in German, I might not have been able to translate it correctly to find what I need.
discrete-mathematics proof-writing relations
edited Jul 26 at 16:00
Bernard
110k635102
asked Jul 26 at 15:09
Awesome36
82
edited Jul 26 at 16:00
Bernard
110k635102
edited Jul 26 at 16:00
Bernard
110k635102
edited Jul 26 at 16:00
Bernard
110k635102
110k635102
asked Jul 26 at 15:09
Awesome36
82
asked Jul 26 at 15:09
Awesome36
82
asked Jul 26 at 15:09
Awesome36
82
82
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
I guess that $R$ left-total means: for every $ain A$, there exists $bin A$ such that $(a,b)in R$.
Suppose $R$ satisfies $mathitId_Asubseteq Rcirc R^-1$; then, for every $ain A$, $(a,a)in Rcirc R^-1$, so there exists $bin A$ with $(a,b)in R$ and $(b,a)in R^-1$.
In particular $(a,b)in R$. Thus $R$ is left-total.
Conversely, suppose $R$ is left-total. Take $ain A$: you want to prove that $(a,a)in Rcirc R^-1$. By assumption, there exists $bin A$ such that $(a,b)in R$.
Then $(a,b)in R$ and $(b,a)in R^-1$, so…
answered Jul 26 at 15:41
egreg
164k1180187
This is not homework I have to do, but it's preparation for my exam and you helped me understanding the proof, so thanks a lot. I would like to ask a couple of questions, if you don't mind: 1.) You assumed what left-total means. I was wondering why I couldn't find anything, that was about left-total Relations. In Germany the property is called 'total'. What would you call it in English? 2.) You proved the if and only if statement by assuming each side and follow the other side. In class we always did: Assume Side A and follow B; Assume not B follow not A. Is there a difference?
– Awesome36
Jul 26 at 15:57
@Awesome36 I've never heard of “left-total†relations; that was just an educated guess. There is no difference in proving $Aimplies B$ and $lnot Bimplies lnot A$. One method for proving $Aiff B$ is proving $Aimplies B$ and $lnot Aimplies lnot B$, but the second part can also be proving that $Bimplies A$.
– egreg
Jul 26 at 16:04
Interesting.. Maybe that's why I could not find anything here, although I assumed the exercise was pretty common. Anyways, thanks a lot for helping me out. Any advice as I'm new to the site?
– Awesome36
Jul 26 at 16:16
@Awesome36 Go forth and practice proofs! :-)
– egreg
Jul 26 at 16:51
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
I guess that $R$ left-total means: for every $ain A$, there exists $bin A$ such that $(a,b)in R$.
Suppose $R$ satisfies $mathitId_Asubseteq Rcirc R^-1$; then, for every $ain A$, $(a,a)in Rcirc R^-1$, so there exists $bin A$ with $(a,b)in R$ and $(b,a)in R^-1$.
In particular $(a,b)in R$. Thus $R$ is left-total.
Conversely, suppose $R$ is left-total. Take $ain A$: you want to prove that $(a,a)in Rcirc R^-1$. By assumption, there exists $bin A$ such that $(a,b)in R$.
Then $(a,b)in R$ and $(b,a)in R^-1$, so…
answered Jul 26 at 15:41
egreg
164k1180187
This is not homework I have to do, but it's preparation for my exam and you helped me understanding the proof, so thanks a lot. I would like to ask a couple of questions, if you don't mind: 1.) You assumed what left-total means. I was wondering why I couldn't find anything, that was about left-total Relations. In Germany the property is called 'total'. What would you call it in English? 2.) You proved the if and only if statement by assuming each side and follow the other side. In class we always did: Assume Side A and follow B; Assume not B follow not A. Is there a difference?
– Awesome36
Jul 26 at 15:57
@Awesome36 I've never heard of “left-total†relations; that was just an educated guess. There is no difference in proving $Aimplies B$ and $lnot Bimplies lnot A$. One method for proving $Aiff B$ is proving $Aimplies B$ and $lnot Aimplies lnot B$, but the second part can also be proving that $Bimplies A$.
– egreg
Jul 26 at 16:04
Interesting.. Maybe that's why I could not find anything here, although I assumed the exercise was pretty common. Anyways, thanks a lot for helping me out. Any advice as I'm new to the site?
– Awesome36
Jul 26 at 16:16
@Awesome36 Go forth and practice proofs! :-)
– egreg
Jul 26 at 16:51
add a comment |Â
up vote
1
down vote
accepted
I guess that $R$ left-total means: for every $ain A$, there exists $bin A$ such that $(a,b)in R$.
Suppose $R$ satisfies $mathitId_Asubseteq Rcirc R^-1$; then, for every $ain A$, $(a,a)in Rcirc R^-1$, so there exists $bin A$ with $(a,b)in R$ and $(b,a)in R^-1$.
In particular $(a,b)in R$. Thus $R$ is left-total.
Conversely, suppose $R$ is left-total. Take $ain A$: you want to prove that $(a,a)in Rcirc R^-1$. By assumption, there exists $bin A$ such that $(a,b)in R$.
Then $(a,b)in R$ and $(b,a)in R^-1$, so…
answered Jul 26 at 15:41
egreg
164k1180187
This is not homework I have to do, but it's preparation for my exam and you helped me understanding the proof, so thanks a lot. I would like to ask a couple of questions, if you don't mind: 1.) You assumed what left-total means. I was wondering why I couldn't find anything, that was about left-total Relations. In Germany the property is called 'total'. What would you call it in English? 2.) You proved the if and only if statement by assuming each side and follow the other side. In class we always did: Assume Side A and follow B; Assume not B follow not A. Is there a difference?
– Awesome36
Jul 26 at 15:57
@Awesome36 I've never heard of “left-total†relations; that was just an educated guess. There is no difference in proving $Aimplies B$ and $lnot Bimplies lnot A$. One method for proving $Aiff B$ is proving $Aimplies B$ and $lnot Aimplies lnot B$, but the second part can also be proving that $Bimplies A$.
– egreg
Jul 26 at 16:04
Interesting.. Maybe that's why I could not find anything here, although I assumed the exercise was pretty common. Anyways, thanks a lot for helping me out. Any advice as I'm new to the site?
– Awesome36
Jul 26 at 16:16
@Awesome36 Go forth and practice proofs! :-)
– egreg
Jul 26 at 16:51
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
I guess that $R$ left-total means: for every $ain A$, there exists $bin A$ such that $(a,b)in R$.
Suppose $R$ satisfies $mathitId_Asubseteq Rcirc R^-1$; then, for every $ain A$, $(a,a)in Rcirc R^-1$, so there exists $bin A$ with $(a,b)in R$ and $(b,a)in R^-1$.
In particular $(a,b)in R$. Thus $R$ is left-total.
Conversely, suppose $R$ is left-total. Take $ain A$: you want to prove that $(a,a)in Rcirc R^-1$. By assumption, there exists $bin A$ such that $(a,b)in R$.
Then $(a,b)in R$ and $(b,a)in R^-1$, so…
answered Jul 26 at 15:41
egreg
164k1180187
I guess that $R$ left-total means: for every $ain A$, there exists $bin A$ such that $(a,b)in R$.
Suppose $R$ satisfies $mathitId_Asubseteq Rcirc R^-1$; then, for every $ain A$, $(a,a)in Rcirc R^-1$, so there exists $bin A$ with $(a,b)in R$ and $(b,a)in R^-1$.
In particular $(a,b)in R$. Thus $R$ is left-total.
Conversely, suppose $R$ is left-total. Take $ain A$: you want to prove that $(a,a)in Rcirc R^-1$. By assumption, there exists $bin A$ such that $(a,b)in R$.
Then $(a,b)in R$ and $(b,a)in R^-1$, so…
answered Jul 26 at 15:41
egreg
164k1180187
answered Jul 26 at 15:41
egreg
164k1180187
answered Jul 26 at 15:41
egreg
164k1180187
answered Jul 26 at 15:41
egreg
164k1180187
164k1180187
This is not homework I have to do, but it's preparation for my exam and you helped me understanding the proof, so thanks a lot. I would like to ask a couple of questions, if you don't mind: 1.) You assumed what left-total means. I was wondering why I couldn't find anything, that was about left-total Relations. In Germany the property is called 'total'. What would you call it in English? 2.) You proved the if and only if statement by assuming each side and follow the other side. In class we always did: Assume Side A and follow B; Assume not B follow not A. Is there a difference?
– Awesome36
Jul 26 at 15:57
@Awesome36 I've never heard of “left-total†relations; that was just an educated guess. There is no difference in proving $Aimplies B$ and $lnot Bimplies lnot A$. One method for proving $Aiff B$ is proving $Aimplies B$ and $lnot Aimplies lnot B$, but the second part can also be proving that $Bimplies A$.
– egreg
Jul 26 at 16:04
Interesting.. Maybe that's why I could not find anything here, although I assumed the exercise was pretty common. Anyways, thanks a lot for helping me out. Any advice as I'm new to the site?
– Awesome36
Jul 26 at 16:16
@Awesome36 Go forth and practice proofs! :-)
– egreg
Jul 26 at 16:51
add a comment |Â
This is not homework I have to do, but it's preparation for my exam and you helped me understanding the proof, so thanks a lot. I would like to ask a couple of questions, if you don't mind: 1.) You assumed what left-total means. I was wondering why I couldn't find anything, that was about left-total Relations. In Germany the property is called 'total'. What would you call it in English? 2.) You proved the if and only if statement by assuming each side and follow the other side. In class we always did: Assume Side A and follow B; Assume not B follow not A. Is there a difference?
– Awesome36
Jul 26 at 15:57
@Awesome36 I've never heard of “left-total†relations; that was just an educated guess. There is no difference in proving $Aimplies B$ and $lnot Bimplies lnot A$. One method for proving $Aiff B$ is proving $Aimplies B$ and $lnot Aimplies lnot B$, but the second part can also be proving that $Bimplies A$.
– egreg
Jul 26 at 16:04
Interesting.. Maybe that's why I could not find anything here, although I assumed the exercise was pretty common. Anyways, thanks a lot for helping me out. Any advice as I'm new to the site?
– Awesome36
Jul 26 at 16:16
@Awesome36 Go forth and practice proofs! :-)
– egreg
Jul 26 at 16:51
This is not homework I have to do, but it's preparation for my exam and you helped me understanding the proof, so thanks a lot. I would like to ask a couple of questions, if you don't mind: 1.) You assumed what left-total means. I was wondering why I couldn't find anything, that was about left-total Relations. In Germany the property is called 'total'. What would you call it in English? 2.) You proved the if and only if statement by assuming each side and follow the other side. In class we always did: Assume Side A and follow B; Assume not B follow not A. Is there a difference?
– Awesome36
Jul 26 at 15:57
This is not homework I have to do, but it's preparation for my exam and you helped me understanding the proof, so thanks a lot. I would like to ask a couple of questions, if you don't mind: 1.) You assumed what left-total means. I was wondering why I couldn't find anything, that was about left-total Relations. In Germany the property is called 'total'. What would you call it in English? 2.) You proved the if and only if statement by assuming each side and follow the other side. In class we always did: Assume Side A and follow B; Assume not B follow not A. Is there a difference?
– Awesome36
Jul 26 at 15:57
@Awesome36 I've never heard of “left-total†relations; that was just an educated guess. There is no difference in proving $Aimplies B$ and $lnot Bimplies lnot A$. One method for proving $Aiff B$ is proving $Aimplies B$ and $lnot Aimplies lnot B$, but the second part can also be proving that $Bimplies A$.
– egreg
Jul 26 at 16:04
@Awesome36 I've never heard of “left-total†relations; that was just an educated guess. There is no difference in proving $Aimplies B$ and $lnot Bimplies lnot A$. One method for proving $Aiff B$ is proving $Aimplies B$ and $lnot Aimplies lnot B$, but the second part can also be proving that $Bimplies A$.
– egreg
Jul 26 at 16:04
Interesting.. Maybe that's why I could not find anything here, although I assumed the exercise was pretty common. Anyways, thanks a lot for helping me out. Any advice as I'm new to the site?
– Awesome36
Jul 26 at 16:16
Interesting.. Maybe that's why I could not find anything here, although I assumed the exercise was pretty common. Anyways, thanks a lot for helping me out. Any advice as I'm new to the site?
– Awesome36
Jul 26 at 16:16
@Awesome36 Go forth and practice proofs! :-)
– egreg
Jul 26 at 16:51
@Awesome36 Go forth and practice proofs! :-)
– egreg
Jul 26 at 16:51
add a comment |Â
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