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 December 9th, 2016, 10:35 AM #1 Newbie   Joined: Nov 2016 From: Texas Posts: 12 Thanks: 0 Equivalence Relations Let S={a,b,c,d,e} and R= {(a,a),(a,c),(a,e),(b,b),(b,d),( c,a), (c,c),(c,e),(d,b),(d,d),(e,a),(e ,c),(e,e)} Is R is reflexive, symmetric, transitive and an equivalence relation ? TRUE/FALSE? My Method/Knowledge/Answer: - Knowledge- Ok so I know the three classes and their rules are: - Reflexive = a~b Symmetric = a~b and b~a Transitive = If a~b and b~c then a~c. Method- For example with R={(a,a)} = This would be Reflexive (Because a is equal to a) R={(a,c)}/{(c,a)} = This would be Symmetric (Because a is equal to b and b is equal to a) R={(a,e)}/{(e,a)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c) R={(c,c)} = This would be Reflexive (Because a is equal to a) R={(c,a)}/{(a,c)} = This would be Symmetric (Because a is equal to b and b is equal to a) R={(c,e)}/{(e,c)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c) R={(e,e)} = This would be Reflexive (Because a is equal to a) R={(e,a)} = This would be Symmetric (Because a is equal to b and b is equal to a) R={(e,c)}/{(c,e)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c) Now this is why I think the whole answer is false is this next part R={(b,b)} = This would be Reflexive (Because a is equal to a) R={(b,d)}/{(d,b)} = This would be Symmetric (Because a is equal to b and b is equal to a) For transitive their would be nothing? Am I right? (Just because their is no transitive here would it make the whole answer false?) FYI: Sorry for the lengthy post just want to put my thoughts to paper, so people dont think I am here for quick answers. December 9th, 2016, 02:48 PM   #2
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 Originally Posted by Mathsisthebestandisamazin Let S={a,b,c,d,e} and R= {(a,a),(a,c),(a,e),(b,b),(b,d),( c,a), (c,c),(c,e),(d,b),(d,d),(e,a),(e ,c),(e,e)} Is R is reflexive, symmetric, transitive and an equivalence relation ? TRUE/FALSE? My Method/Knowledge/Answer: - Knowledge- Ok so I know the three classes and their rules are: - Reflexive = a~b Symmetric = a~b and b~a Transitive = If a~b and b~c then a~c. Method- For example with R={(a,a)} = This would be Reflexive (Because a is equal to a) R={(a,c)}/{(c,a)} = This would be Symmetric (Because a is equal to b and b is equal to a) R={(a,e)}/{(e,a)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c) R={(c,c)} = This would be Reflexive (Because a is equal to a) R={(c,a)}/{(a,c)} = This would be Symmetric (Because a is equal to b and b is equal to a) R={(c,e)}/{(e,c)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c) R={(e,e)} = This would be Reflexive (Because a is equal to a) R={(e,a)} = This would be Symmetric (Because a is equal to b and b is equal to a) R={(e,c)}/{(c,e)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c) Now this is why I think the whole answer is false is this next part R={(b,b)} = This would be Reflexive (Because a is equal to a) R={(b,d)}/{(d,b)} = This would be Symmetric (Because a is equal to b and b is equal to a) For transitive their would be nothing? Am I right? (Just because their is no transitive here would it make the whole answer false?) FYI: Sorry for the lengthy post just want to put my thoughts to paper, so people dont think I am here for quick answers.
For R to be reflexive we must have that {x,x} is in R. Since we have {a,a}, {b,b}, {c,c}, {d,d}, {e,e} in R we can say R is reflexive. We can't say that {a,a} is reflexive because this is just a single element of R.

The same goes for symmetry: If we have {x, y} we must have {y, x} in R. We don't have to have all such pairs but again we are looking for a subset of R such that this is true. We can't say that a~c is reflexive...this is only one element in the set defining R.

The same kind of thing goes for transitive. We can't say that {c, e}, {e, c} in R defines a transitive element without also having an element b such that {c, b} and {b, d} and {c, d} are all in R.

Your R is reflexive and symmetric. Take another look at the transitive property: If we have {a, c} and {c, e} then we must also have {a, e}. We have {b, d} and {d, e}. Do we have {b, e}? etc.

To answer your direct question, just because we have {b, d} in R does not mean that there is a transitive element involved, as long as we don't have any elements (d, x) and no element (b, x). But it can have the element {b, d} alone.

-Dan

Last edited by topsquark; December 9th, 2016 at 02:50 PM. Tags equivalence, relations Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Jet1045 Abstract Algebra 5 May 19th, 2013 01:03 PM bvh Advanced Statistics 2 April 10th, 2013 09:18 PM shine123 Applied Math 1 March 5th, 2013 01:52 AM remeday86 Applied Math 1 June 13th, 2010 12:10 AM shine123 Number Theory 1 December 31st, 1969 04:00 PM

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