I'm teaching a discrete math class at the high school level and realize that I'm fuzzier on a topic than I should be.

In their last problem set, I asked my students to translate "There is a triangle that is above every square." into formal notation.

My answer was $\exists t\forall s(\text{triangle}(t)\wedge\text{square}(s)\rightarrow\text{above}(t, s))$.

Many students made a mistake of replacing the implication with conjunction, but once I explained they were saying that all members of the domain were squares, they understood why they needed the implication.

A few students side-stepped the problem by writing $\exists t\ \text{triangle}(t)(\forall s\ \text{square}(s)(\text{above}(t, s)))$. This was inspired by a quasi-formal notation in the textbook that would have written the sentence as $\exists\ \text{triangle}\ t,\ \forall\ \text{squares}\ s,\ \text{above}(t, s)$.

What I'm not sure about is whether this is okay. Is there some notational convention that $$ \forall xP(x)(Q(x))\equiv\forall x(P(x)\rightarrow Q(x))$$ or am I just making that up?

What confuses this somewhat is that I'm pretty sure that $$\exists xP(x)(Q(x))\equiv\exists x(P(x)\wedge Q(x))$$ under the normal understanding of the notation without any convention, so I'm afraid that I let my students get away with playing fast and loose with the notation when I shouldn't have.

Thanks! Todd