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 for all students in your class.
- 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.∃x∀y((x $\neq$ y) → (¬M(y, x) ∧ ¬T(y, x)))
- Every student in the class has either received an e-mail message or received a telephone call
from another student in the class.∀x∃y((x $\neq$ y) ∧ (M(y, x) ∨ T(y, x)))
I can't understand why in the first quantifier we used the → mark and did not directly use the ∧ mark, and if so why didn't use the → mark in the second one. I can't see why should we use different symbols because the sentences that needs to be translated do not show a reason why the first is an implication mark and the second is the conjunction mark after the (x $\neq$ y)
Best Answer
For the 1st sentence
The difference between using conjunction or implication in the first sentence becomes clearer if you consider the two versions of the sentence for a class with only one student. Suppose her name is An.
In this class, $$\exists x\forall y((x\neq y) \land(\neg M(y,x)\land\neg T(y,x)))$$ is false, because there is no $y$ such that $(x\neq y)$. $x$ necessarily is An, and if there is some $y$ such that $(\neg M(y,x)\land\neg T(y,x))$, then necessarily $(y = \text{An})$ too. So, $(x=y)$ and this falsifies the sentence.
On the other hand, $$\exists x\forall y((x\neq y) \to(\neg M(y,x)\land\neg T(y,x)))$$ is true, because it treats $(x \neq y)$ only as a condition and not as a fact that must hold. It says that "if there is a student other than An, then [...]". In this class, it happens that there is no student other than An. Since the condition is not met, the sentence is true no matter the conclusion of the implication.
This question is related and possibly helpful.
For the 2nd sentence
Consider the two versions of the sentence: $$(\forall x\exists y((x\neq y)\to(M(y,x)\lor T(y,x))))~~\text{and}~~(\forall x\exists y((x\neq y)\land(M(y,x)\lor T(y,x)))).$$ What are the truth values of the two sentences, in a class where everyone has sent an email only to themselves and to no-one else?