I studied set theory in high school, but I'm seeing a period in my university course, that I can't recognize to mean something. Here is an example from my professor's slides:
The notation $\{ (x, y) \mid Φ \}$ abbreviates $\{ p \mid ∃x. ∃y. p = (x, y) ∧ Φ \}.$
The sentence $$∃x\; ∃y\; p = (x, y) ∧ Φ(x,y)\tag0$$ conventionally means $$\Big[∃x\; ∃y\; p = (x, y) \Big]∧ Φ(x,y)\tag1$$ $$\Big[∃a\; ∃b\; p = (a, b) \Big]∧ Φ(x,y).\tag1$$ If we add periods to sentence $(0)$ in the way that your lecturer does, then there is often ambiguity whether we are inserting them in a manner that it logically superfluous (for example, just to add visual space), OR whether we are using them as delimiters to specify sentence $(0)$'s intended meaning as $$∃x\; ∃y\; \Big[p = (x, y) ∧ Φ(x,y)\Big].\tag2$$ Note that sentences $(1)$ and $(2)$ are not equivalent to each other.
The notation $\{ (x, y) \mid Φ \}$ abbreviates $\{ p \mid ∃x. ∃y. p = (x, y) ∧ Φ \}.$
It turns out that your lecturer actually means sentence $(2)$ instead of sentence $(1),$ since the notation $$\left\{ (x, y) \mid Φ(x,y) \right\}$$ abbreviates $$\left\{ p \;\middle|\; ∃x\;∃y\; \Big[p = (x, y) ∧ Φ(x,y) \Big] \right\}.$$ (Notice that they had neglected to indicate that $Φ$ contains the free variables $x$ and $y.$) In other words, that slide is incorrect unless we read those periods as indicating that those existential quantifiers' scope extends beyond that ∧ symbol.
Please click on the second link for a fuller answer regarding the use of periods in formal sentences; in short, though: avoid them, or use them only with a prefacing usage note so that readers will be sure to read your logic formulae in your intended way.
