[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.)