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Please see the question which shows you two questions and two answers. Are the 2 answers contradicting each other? Can someone clarify please.

Relations quesion

Question 1 Answer says "DOES NOT IMPLY THAT ALL X and Y...." but Question 4 answers says "IMPLIES FOR ALL X and Y

Sanone
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  • Please type out all images. Images are not searchable and some users will get a bad impression, among other issues. Formatting tips here. – Em. May 08 '17 at 08:40
  • The first red underlined question of the OP is the same one as he has asked half an hour ago in (https://math.stackexchange.com/q/2271144) but he/she doesn't mention it !!! This question should be closed. – Jean Marie May 08 '17 at 08:53
  • Read again. They are not same. – Sanone May 08 '17 at 09:15

2 Answers2

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The definition of symmetric relation is:

for every $a,b \in X \ (aRb \Leftrightarrow bRa)$.

This means that, for every pair of element $a,b$: either $a,b$ and $b,a$ are $R$-related or neither $a,b$ nor $b,a$ are.

If $aRb \Leftrightarrow bRa$, obviously $aRb \Rightarrow bRa$, and this explain Q4's answer: if $R$ is symm, then $xRy$ implies $yRx$, for every $x,y$.

But the def of symmetry does not mean that every pair $a,b$ must be $R$-related.

This explain why it is not true that $xRy$ and $yRx$.

Consider the trivial example with the symmetric relation: $\text { is brother of }$.

Obviously, if $\text {John is brother of Jim }$, then $\text {Jim is brother of John }$, but this does not mean that every two men in the world are brothers (at least in the "biological" sense...).

In more formal terms, we have that form the definition of symmetry we derive correctly that: $∀x \ ∀y \ (xRy → yRx)$.

From the wrong answer above, insted, we derive: $∀x \ ∀y \ (xRy)$, that is not licensed by the definition of symmetry.

Mauro ALLEGRANZA
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  • Thanks Mauro. I really like the brother analogy. I am from a non native English speaking country. The sentence Relation R on a set X is symmetric, then xRy and yRx for all x,y E X and R to be symmetric on X, we have that if xRy for x,y E X then yRx also have two different meanings I guess. – Sanone May 08 '17 at 09:04
  • @Sanone - two aspects; one is related to propositional logic: $p \to q$ (if xRy, then yRx) is not the same as $p \land q$ (xRy and yRx). – Mauro ALLEGRANZA May 08 '17 at 10:29
  • The other is the def of symm: R is symm iff: for all $x,y \in X$ (xRy iff yRx). From this we have that: if R is symm, then xRy iff yRx and thus (the iff is a bi-conditional, i.e. the conjunctions of the two separated conditionals) if R is symm, then if xRy, then yRx. – Mauro ALLEGRANZA May 08 '17 at 10:31
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In both answers a symmetric relation on $Y$ is described as a relation that has the property: $$xRy\implies yRx\text{ for all }x,y\in Y\tag1$$ It is evidently not true that $(1)$ implies that $xRy$ for every pair $x,y\in Y$.

This fact is mentioned in the first answer (where $Y=X$) as some extra information, and is not mentioned in the second answer (where $Y=S$).

drhab
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