Consider that statement: "There is math course that no first-year student taking." Let $A(y)$, $B(x)$ and $C(x, y)$ be the predicates "$y$ is a math course", "$x$ is a first-year student" and "$x$ is taking course $y$", respectively. Consider that the domain of $x$ and $y$ is the universe of all students and all the courses, respectively.
I know that "There is math course that no first-year student taking." can be expressed as $\exists y\forall x (A(y) \wedge (B(x) \rightarrow\neg C(x, y)))$.
Can this same statement be expressed as $\exists y\neg \exists x (A(y) \wedge B(x) \wedge C(x, y))$? I think that this makes sense but these to expressions are not equivalent after applying De Morgan's law.
Best Answer
Notice, from the specific way that you've set up the predicates, that separate discourse domains are unnecessary (but harmless) here.
Statement $(2)$ translates to “There is a course for which, for no student, the course is math and the student is in year 1 and taking it”. It is strictly weaker than statement $(1),$ i.e., $$(1)\implies(2),\\(2)\kern.6em\not\kern-.6em\implies(1).$$ For example, $(2)$ is automatically true if the institution offers a non-mathematics course.