Express statements using symbolic logic

Question

Consider the predicates

$M(x,y):$ "x has sent an email to y",

$T(x,y):$ "x has called y".

The predicate variable x, y take values in the domain D = {students in the class}. I need to express these statements using symbolic logic:

1. "There are at least 2 students in the class such that one student has sent the other an email, and the second student has called the first student." (I don't know how to translate this using symbolic logic.)

2. "There are some students in the class who have emailed everyone": $\exists x\in D, \forall y\in D M(x, y)\quad$?

Answer 1

$$(1)\quad \exists x \exists y\Big( x \in D \land y \in D \land x\neq y \land M(x, y) \land T(y, x)\Big)$$

Alternatively, $$\exists x \in D,\;\exists y \in D\Big(x\neq y \land M(x, y) \land T(y, x)\Big)$$

Note: We need $x \neq y$ to ensure we are talking about at least two students in the class.

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$$(2)\quad \exists x\Big(x \in D \land \forall y(y\in D \rightarrow M(x, y)\Big)$$

Alternatively: $$\exists x \in D,\; \forall y \in D\;(M(x, y))$$