How would one write "There are teachers that teach students" using quantifiers?

Question

Regarding the statement "There are teachers that teach students" I'm unsure on how to properly write it as a proposition. The way I wrote it in my notes was:

Let: I(x) : x is a teacher

S(y) : y is a student

T(x, y) : x is teaching y

followed by:

∃x(I(x) ∧ T(x, ∃xS(y))

my logic being as "There are some teachers AND they teach some students", but I am very unsure if I am allowed to use ∃xS(y) in the middle of the proposition. Or is the ∃x quantifier before the S(y) superfluous and it would be correct to only write "∃x(I(x) ∧ T(x, S(y))"? Have I done it correctly or is there a different way one is meant to do it?

Answer 1

Regarding the statement "There are teachers that teach students"

Let: I(x) : x is a teacher

S(y) : y is a student

T(x, y) : x is teaching y

$∃x(I(x) ∧ T(x, ∃xS(y))$

Correction: $$∃x\,∃y\,(Ix∧Sy∧Txy).$$

Analogously, there are even numbers that are also multiples of 3: $$∃x\,∃y\,(Ex∧My∧x=y).$$

Answer 2

I don't think you should use $\exists x$ inside of your expression, as quantifiers are usually put in front of an expression. So I'd make it $\exists x\exists y:I(x)\land T(x,y)$. However this says that there is $\textbf{a}$ teacher teaching a student, whereas your question asks for multiple teachers.

Answer 3

Could you just define a set that contains all teachers who are teaching students and assert the size of the set is greater than 1?

$|\{ x | \exists y (I(x)\land S(y) \land T(x, y))\}| > 1$

Answer 4

So I think that most people here are almost entirely correct, but when I took discrete, I remember it being important that you specify that the teacher and the student are not the same person:

∃x∃y(I(x)^S(x)^T(x,y)^x!=y)