[Artur Grzesiak wrote:](https://forum.azimuthproject.org/discussion/comment/16937/#Comment_16937)

> There is a difference in defining symmetry.

> In the lecture: for all \\(x,y \in X\\), \\(x \sim y\\) implies \\(y \sim x.\\)

> In the book (definition 1.8): \\(a \sim b\\) iff \\(b \sim a\\), for all \\(a,b \in X\\)

> Implication is weaker than *iff*, but to be honest I am not sure if this is the root cause of the problem, as for me they *feel*... *equivalent* in this particular case.

As you note, the book's definition implies mine, because "P iff Q" implies "P implies Q".

**Puzzle.** Prove that my definition implies the book's definition.

(I have changed the set \\(A\\) in your comment to \\(X\\), so that the two definitions are talking about the same set.)

> There is a difference in defining symmetry.

> In the lecture: for all \\(x,y \in X\\), \\(x \sim y\\) implies \\(y \sim x.\\)

> In the book (definition 1.8): \\(a \sim b\\) iff \\(b \sim a\\), for all \\(a,b \in X\\)

> Implication is weaker than *iff*, but to be honest I am not sure if this is the root cause of the problem, as for me they *feel*... *equivalent* in this particular case.

As you note, the book's definition implies mine, because "P iff Q" implies "P implies Q".

**Puzzle.** Prove that my definition implies the book's definition.

(I have changed the set \\(A\\) in your comment to \\(X\\), so that the two definitions are talking about the same set.)