The difference between “if” and “and” in symbolic logic

Question

Let $M(x, y)$ be “$x$ has sent $y$ an e-mail message” and
$T (x, y)$ be “$x$ has telephoned $y,$” where the domain consists
of all students in your class. Assume that all e-mail
messages that were sent are received.
Use quantifiers to express each of these statements.

1. There is a student in your class who has not received
   an e-mail message from anyone else in the class and
   who has not been called by any other student in the
   class.

2. Every student in the class has either received an email
   message or received a telephone call from another
   student in the class.

The answer in book for 1 is: $$∃x∀y(x ≠ y → (¬M(y, x) ∧¬T (y, x)))$$
and the answer for 2 is: $$∀x(∃y(x ≠ y ∧ (M(y, x) ∨ T (y, x)))).$$

My problem is with parts $$x ≠ y\: ∧$$ and $$x ≠ y →.$$ When should I use "→" or "∧" after $x ≠ y?$ I don't understand the difference in their literature in the question. When I want to use "if" in any other question, the answer is "and" and vice versa.

Answer 1

1. • “There is a student who is furry and violet and handsome” is a simple statement of three properties of a particular existent (albeit unnamed) student $$\exists s \;\Big(Fs \land \big(Vs \land Hs\big)\Big);$$ because there is nothing hypothetical or conditional about this statement (the student's colour and looks are not dependent on their furriness), replacing the first $\land$ with $\to$ will muck up the translation.
   • “Every furry student is violet and handsome”, on the other hand, states that if a student is furry, then they must also be violet and handsome $$\forall s \;\Big(Fs \to \big(Vs \land Hs\big)\Big);$$ notice that the statement makes no claim about non-furry students, or that furry students even exist. If $\to$ is replaced with $\land,$ the meaning will become “Every student is furry and violet and handsome”.
2. • In your second exercise $$∀x∃y\;\Big(x{≠}y ∧ \big(M(y, x) ∨ T (y, x)\big)\Big),$$ we are merely exhibiting two properties of $y$ (who is claimed to exist), the first property being that $y$ is distinct from $x.$
   • In your first exercise $$∃x∀y\;\Big(x{≠}y → \big(¬M(y, x) ∧¬T (y, x)\big)\Big),$$ we are exhibiting a property of $y\:$ for the case where $\mathbf y$ is distinct from $\mathbf x.$ Changing the → to ∧ will, by additionally asserting that no student ever emails or calls themself, result in a stronger statement than was given.