# Express statements using symbolic logic

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": $\exists x\in D, \forall y\in D M(x, y)\quad$?

• No idea ? For the second one you have to use both quantifiers $\exists$ and $\forall$. For the first one you have to use two occurrences of $\exists$, but take care of the condition : "at least 2 students "... – Mauro ALLEGRANZA Sep 5 '14 at 12:50
• I have no clue on how to take care if it checks for more than 1. For the 2nd one I came out with the answer ∃x∈D ∀y∈D, M(x,y) – Gavin Sep 5 '14 at 12:53
• The second is Ok; for the first one, you have to start with : $\exists x \exists y ( x \ne y \ldots)$. – Mauro ALLEGRANZA Sep 5 '14 at 12:54

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

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.

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

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

• Interesting how does x ≠ y ensure that there are at least 2 students? Could you help me out with that? – Gavin Sep 5 '14 at 13:09
• If we don't eliminate the possibility that $x = y$, then the statement would be true even if there is only one student who happens to email himself and then to call himself. But requiring $x \neq y$, then there is at least one $x\neq y$ who emails y, and at least one $y$, different from x, who calls $x$. It would get more complicated if we required two and only two such students, but all we need to ensure for this translation is "at least two students", which can be guaranteed by requiring $x\neq y$ – Namaste Sep 5 '14 at 13:13