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 Post subject: HELP!! Relation xRx problemPosted: Fri, 30 Mar 2012 20:43:56 UTC
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Joined: Fri, 30 Mar 2012 20:32:55 UTC
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Consider the proposition: :et R be a relation on a set X. If R is symmetric and transitive, then it is reflexive. The proposition is false. However, someone wrote the following proof. What is wrong with it.

Proof: "Let x and y be elements of X. If xRy, then yRx since R is symmetric. Then, by transitivity, we have that xRx, so that R is reflexive. QEd."

My answer (wrong): I said that in the proposition, it says: "If R symmetric AND transitive",
but in the proof, it is : "if R symmetric, then transitive".

so in the proof, it is "then" instead of "and".

Can someone explain to me what else is wrong with my answer. thank you!!

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 Post subject: Re: HELP!! Relation xRx problemPosted: Fri, 30 Mar 2012 21:18:57 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
GoldenPho wrote:
Consider the proposition: :et R be a relation on a set X. If R is symmetric and transitive, then it is reflexive. The proposition is false. However, someone wrote the following proof. What is wrong with it.

Proof: "Let x and y be elements of X. If xRy, then yRx since R is symmetric. Then, by transitivity, we have that xRx, so that R is reflexive. QEd."

My answer (wrong): I said that in the proposition, it says: "If R symmetric AND transitive",
but in the proof, it is : "if R symmetric, then transitive".

so in the proof, it is "then" instead of "and".

Can someone explain to me what else is wrong with my answer. thank you!!

What if R is the empty relation?

More completely: your proof assumes that there is always x and y which are related in the first place, and that's where the error lies. If there is some element unrelated to any other element, then you cannot apply this proof which assumes the existence of a y related to x. However symmetry and transitivity do not give any kind of existence to you.

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 Post subject: Re: HELP!! Relation xRx problemPosted: Sat, 31 Mar 2012 00:35:28 UTC
 S.O.S. Newbie

Joined: Fri, 30 Mar 2012 20:32:55 UTC
Posts: 4
Thank you for the help!!

So if I understand correctly, what is wrong in the proof is the assumption that "x and y in X (a certain set)" are related to each other.

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 Post subject: Re: HELP!! Relation xRx problemPosted: Sat, 31 Mar 2012 00:56:06 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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Location: Austin, TX
GoldenPho wrote:
Thank you for the help!!

So if I understand correctly, what is wrong in the proof is the assumption that "x and y in X (a certain set)" are related to each other.

Yes, because there might not be ANY y related to a given x.

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