Does Such a Relation Exist?
in [Logic]
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From: Charlie-Boo on 5 Nov 2009 10:02 Is there a two-place relation R such that: 1. If x is an element of y then there exists a z such that R(y,z). 2. If R(x,y) then y is an element of x. 3. If R(x,y) and R(x,z) then y=z. What should it be called? C-B
From: James Burns on 5 Nov 2009 10:29 Charlie-Boo wrote: > Is there a two-place relation R such that: > > 1. If x is an element of y then there exists a z such that R(y,z). > > 2. If R(x,y) then y is an element of x. > > 3. If R(x,y) and R(x,z) then y=z. > > What should it be called? Would you mind sharing why the question is interesting? (2) and (3) imply that R(x,y) iff x = {y} and (1) certainly follows from that (with z = x). So, yes to your first question; there is such a relation. I'll leave it to you to come up an answer to your second question, that is, a name. Jim Burns
From: James Burns on 5 Nov 2009 10:43 James Burns wrote: > Charlie-Boo wrote: > >> Is there a two-place relation R such that: >> >> 1. If x is an element of y then there exists a z such that R(y,z). >> >> 2. If R(x,y) then y is an element of x. >> >> 3. If R(x,y) and R(x,z) then y=z. >> >> What should it be called? > > > Would you mind sharing why the question is interesting? > > (2) and (3) imply that > > R(x,y) iff x = {y} > > and (1) certainly follows from that (with z = x). Ooops. (1) only follows from (2) and (3) if y is of the correct type, that is, if y is a singleton (or empty). So, in order for such a relation to exist, its range on its first parameter needs to be restricted to singletons (and the empty set). This wasn't specified in the problem statement, but, if the unspecified sets X and Y for which R is a subset of XxY also satisfy X = Y, then I think X = { {}, {{}}, {{{}}}, {{{{}}}}, ... } is the unique solution (err, that and {} ). > > So, yes to your first question; there is such a relation. > > I'll leave it to you to come up an answer > to your second question, that is, a name. > > Jim Burns
From: Charlie-Boo on 5 Nov 2009 13:45 On Nov 5, 10:29 am, James Burns <burns...(a)osu.edu> wrote: > Charlie-Boo wrote: > > Is there a two-place relation R such that: > > > 1. If x is an element of y then there exists a z such that R(y,z). > > > 2. If R(x,y) then y is an element of x. > > > 3. If R(x,y) and R(x,z) then y=z. > > > What should it be called? > > Would you mind sharing why the question is interesting? A lot of people write about it. > (2) and (3) imply that > > R(x,y) iff x = {y} Not quite e.g. R(x,y) iff x = {y,{}} also satisfies (2) and (3). > and (1) certainly follows from that (with z = x). See your next post. > So, yes to your first question; there is such a relation. > > I'll leave it to you to come up an answer > to your second question, that is, a name. How about answer to a long standing problem? C-B > Jim Burns
From: Charlie-Boo on 5 Nov 2009 13:47
On Nov 5, 10:43 am, James Burns <burns...(a)osu.edu> wrote: > James Burns wrote: > > Charlie-Boo wrote: > > >> Is there a two-place relation R such that: > > >> 1. If x is an element of y then there exists a z such that R(y,z). > > >> 2. If R(x,y) then y is an element of x. > > >> 3. If R(x,y) and R(x,z) then y=z. > > >> What should it be called? > > > Would you mind sharing why the question is interesting? > > > (2) and (3) imply that > > > R(x,y) iff x = {y} > > > and (1) certainly follows from that (with z = x). > > Ooops. (1) only follows from (2) and (3) if y is > of the correct type, that is, if y is a singleton > (or empty). So, in order for such a relation > to exist, its range on its first parameter needs > to be restricted to singletons (and the empty set). > > This wasn't specified in the problem statement, but, > if the unspecified sets X and Y for which R is > a subset of XxY also satisfy X = Y, then > I think X = { {}, {{}}, {{{}}}, {{{{}}}}, ... } > is the unique solution Are you saying that X is a Quine atom (to satisfy 2)? > (err, that and {} ). R cant be {} because (1) needs some tuples in R. C-B > > > > So, yes to your first question; there is such a relation. > > > I'll leave it to you to come up an answer > > to your second question, that is, a name. > > > Jim Burns- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - |