Formalize the following english statements as quantified WFF.
Let S(x) be the predicate "x is a student," B(x) the predicate "x is a book," and H(x,y) the predicate "x has y, " where the universe of discourse is the universe, that is the set of all objects. Use quantifiers to express each of the following statements.
a. Every student has a book.
b. Some student does not have any book.
c. Some student has all the books.
d. Not every student has a book.
e. There is a book which every student has.
a. ∀x(S(x) → ∃y(B(y) ∧ H(x,y)))
b. ∃x(S(x) ∧ ¬∃y(B(y) ∧ H(x,y)))
c. ∃x(S(x) ∧ ∀y(B(y) → H(x,y)))
d. ¬∀x(S(x) → ∃y(B(y) ∧ H(x,y)))
e. ∃y(B(y) ∧ ∀x(S(x) → H(x,y)))
a. ∀x(S(x) → ∃y(B(y) ∧ H(x,y)))
b. ∃x(S(x) ∧ ¬∃y(B(y) ∧ H(x,y)))
c. ∃x∃y(S(x) ∧ B(y) ∧ ∀z(B(z) → H(x,z)))
d. ¬∀x(S(x) → ∃y(B(y) ∧ H(x,y)))
e. ∃y(B(y) ∧ ∀x(S(x) → H(x,y)))
To formalize the given English statements as quantified Well-Formed Formulas (WFF), we can use the following quantifiers:
∀: The universal quantifier "for all" or "every"
∃: The existential quantifier "there exists" or "some"
Using these quantifiers and the given predicates S(x), B(x), and H(x, y), we can express each statement as follows:
a. Every student has a book.
∃x (S(x) → ∃y B(y) ∧ H(x, y))
This statement asserts that there exists an x such that if x is a student, then there exists a y such that y is a book and x has y.
b. Some student does not have any book.
∃x (S(x) ∧ ¬∃y (B(y) ∧ H(x, y)))
This statement asserts that there exists an x such that x is a student, and there does not exist a y such that y is a book and x has y.
c. Some student has all the books.
∃x (S(x) ∧ ∀y (B(y) → H(x, y)))
This statement asserts that there exists an x such that x is a student, and for all y, if y is a book, then x has y.
d. Not every student has a book.
¬∀x (S(x) → ∃y B(y) ∧ H(x, y))
This statement asserts that it is not the case that for every x, if x is a student, then there exists a y such that y is a book and x has y.
e. There is a book which every student has.
∃y (B(y) ∧ ∀x (S(x) → H(x, y)))
This statement asserts that there exists a y such that y is a book, and for every x, if x is a student, then x has y.