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February 4th, 2019, 01:42 PM  #11  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 311 Thanks: 109 Math Focus: Number Theory, Algebraic Geometry  Quote:
So, I'll repeat the counterexample once more: 4 is related to 2 (i.e. (4,2) is in R) and 2 is related to 1 (i.e. (2,1) is in R) but 4 is not related to 1 (i.e. (4,1) is not in R). What a surprise...  
February 4th, 2019, 01:43 PM  #12 
Global Moderator Joined: Dec 2006 Posts: 20,633 Thanks: 2080 
The definition you gave, zylo, is right and uses "then" (meaning "it's implied that") just once. In contrast, you used "implies" three times. What you did was different from what you should have done.

February 5th, 2019, 07:27 AM  #13  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 
The problem with thread is the absence of a hard and fast unequivocal definition, leading to inanities like "You are wrong." A binary Relation on a set U is a set S of pairs (a,b) of members of U. The Relation is transitive if (a,b), (b,c) and (a,c) are in S. cjem's example: (4,2) and (2,1) are in S. (4,1) isn't. EDIT: cjem came close. Quote:
Last edited by zylo; February 5th, 2019 at 07:41 AM.  
February 5th, 2019, 08:33 AM  #14 
Global Moderator Joined: Dec 2006 Posts: 20,633 Thanks: 2080  The use of an implication is necessary, not misleading, as it's intended that R might be transitive even though (a, b) and (a, c) are in R, but (b, c) isn't.
Last edited by skipjack; February 5th, 2019 at 08:44 AM. 
February 5th, 2019, 08:37 AM  #15  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 311 Thanks: 109 Math Focus: Number Theory, Algebraic Geometry  Quote:
Quote:
The definition you've given here is much too restrictive: it would mean the only transitive relation on a set U is the set of all pairs (a,b) of elements of U. (Indeed, suppose S is a relation on U such that (a,b) is not in S for some a,b in U. Then, for example, taking c = a, it is not true that "(a,b), (b,c) and (a,c) are in S".) There is an ifthen/implies in the definition, and it's crucial.  
February 5th, 2019, 08:59 AM  #16 
Global Moderator Joined: Dec 2006 Posts: 20,633 Thanks: 2080  That doesn't hold if you are referring to the left and right side of different elements of R. If b appears on both sides of the same element of R, it's trivially true that the element is (b, b). In either case, "which it isn't" isn't known.

February 5th, 2019, 11:37 AM  #17  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
As for implication, what does that mean? it just muddies the water. It implies causality. (a,c) is not in S because (a,b) and (b,c) are. It is there by definition of the relationship, and the test of transitivity is not causality, it is the above definition. Ifthen in this case is pseudomathematics. It is trying to combine a concept from logic with a concept from mathematics. Does (2<3) and (3<4) imply (2<4)? No, (2<4) is in your set of defined relations. You are confusing implication with definition of the relationship.  
February 5th, 2019, 12:14 PM  #18  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 311 Thanks: 109 Math Focus: Number Theory, Algebraic Geometry  Quote:
1) Suppose A is a false statement and B is any statement. Then A implies B. 2) Suppose C is any statement and D is a true statement. Then C implies D. Back to the problem at hand: even though 2 being less than 3 and 3 being less than 4 might not "cause" 2 to be less than 4, it does imply it. Indeed, take C to be the statement "2 < 3 and 3 < 4" and D to be the statement "2 < 4". Since D is true, we see that C implies D (as in example 2) above). Last edited by cjem; February 5th, 2019 at 12:47 PM.  
February 6th, 2019, 07:05 AM  #19 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 
Personally I find it easier to just observe (2,3), (3,4), and (2,4) are in the set. It works for OP without having to think do A and B imply C, and for the order relation among integers. And removes ambiguity of the extra definition. 
February 6th, 2019, 10:18 AM  #20  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 311 Thanks: 109 Math Focus: Number Theory, Algebraic Geometry  Quote:
But perhaps it's best to use the standard definition, or one that's equivalent to it. Here's an "implicationfree" definition that is equivalent to the standard one: A relation U on a set S is transitive if: for all a,b,c in U, (a,c) is in S or (a,b) is not in S or (b,c) is not in S.  

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