$L(x,y)$ be "x eats lunch with y." $C(x,y)$ be "x has class with y." $R(x,y)$ be "x is roommates with y." The domain for x and y is the students at a university. Translate the following into English:
b) $\exists x \forall y(((x \ne y) \land C(x,y))\rightarrow\lnot L(x,y))$
My Answer: For each student, if they are who is not themself, and has class with every student, then this student doesn't live with every student.
c) $\forall x \exists y ((x \ne y) \land ((C(x,y) \lor R(x,y)) \land \lnot L(x,y)) $
My Answer: Every student, who is not themself, and every student has class with this one student or lives with this one student, and every student doesn't live with this student.
My problem with this translation is I don't know if I am supposed to say for $x \ne y$ that these students are not themselves or they're unique?
Also, how to use English for $\exists$ and $\forall$. Some helpful tips and double checking to see if my English is right would be greatly apprecaited!
Best Answer
Use these:
So then the first sentence becomes "There is a student such that any other student who has a class with the first will not eat lunch with the first student". Or, in a simplified version, "There is a student who does not eat lunch with anyone else in his class"