Relations question - need clarification to understand better

Question

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

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

Answer 1

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 $yRx$.

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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...).

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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.

Answer 2

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 $Y=S$).