Writing this statement with quantifiers

Question

Consider the statement "There is a math class that every first-year student is taking". Let $x$ represent a student and $y$ a class. Let $A(y)$ be "y is a math class", $F(x)$ be "x is a first-year student" and $T(x, y)$ be $x$ is taking class $y$. Consider the universe of all student and classes. I know that the statement can be written as $$ \exists y \forall x: A(y) \wedge(F(x) \rightarrow T(x, y)).$$ Why I cannot write it as $ \exists y \forall x: A(y) \wedge F(x) \wedge T(x, y)$?

Answer 1

$$\exists y \forall x \Big(A(y) \wedge F(x) \wedge T(x, y)\Big)\tag1$$ logically implies $$\forall x F(x),\tag2$$ that is, that every object in the universe is a first-year student.

Since the given statement does not imply sentence $(2),$ its formalisation cannot be sentence $(1).$

Answer 2

The two statements you have in symbolic form aren't logically equivalent. If we're supposing $$\exists y \forall x: A(y) \wedge(F(x) \rightarrow T(x, y))$$ is true, then the equivalent statement is $$\exists y \forall x: A(y) \wedge (\neg F(x) \vee T(x, y)).$$

Generally speaking, $P \implies Q$ is equivalent to $\neg P \vee Q$, and you can verify that using truth tables.