I have come across few doubts solving Exercise of Propositional logic and predicates. Here are they,
Doubt 1
Let T (x, y) mean that student x likes cuisine y, where the
domain for x consists of all students at your school and
the domain for y consists of all cuisines. Express each of
these statements by a simple English sentence.
e) ∃x∃z∀y(T (x, y) ↔ T (z,y))
f ) ∀x∀z∃y(T (x, y) ↔ T (z,y))
Solutions:
e) There are two students at your school who like exactly the same set of cuisines.
f) For every pair of students at your school, there is some cuisine about
which they have the same opinion (either they both like it or they both do not like it).
My doubt is can i write for e) There are two students at your school x and y such that x likes a cuisine iff y likes it. f) For Every pair of students at your school x and y, x likes a cuisine iff y likes it. Am i correct in saying that, that the only difference between them is about "two students" and "every pair" or there's any other as being cited by the Textbook Solutions. Can you please elaborate what textbook solutions are implying?
Doubt 2
Let S(x) be the predicate “x is a student,” F(x) the predicate
“x is a faculty member,” and A(x, y) the predicate
“x has asked y a question,” where the domain consists of
all people associated with your school. Use quantifiers to
express each of these statements.
f ) Some student has asked every faculty member a question.
Solution: f) ∀y(F(y)→∃x(S(x)∨A(x,y)))
Shouldn't the solution be ∃x(S(x)∧∀y(F(y)→A(x,y))) ?
Doubt 3
Express each of these system specifications using predicates,
quantifiers, and logical connectives, if necessary.
∗d) There are exactly two systems that monitor every remote
server.
How to express it?