**Unit 8 Exercises**

**1. **Let *L(x, y)* be the predicate "*x* likes y,"
and let the universe of discourse be the set of all people. Use quantifiers to express
each of the following statements.

a) Everyone likes everyone.

b) Everyone likes someone.

c) Someone does not like anyone.

d) Everyone likes George.

e) There is someone whom everyone likes.

f) There is no one whom everyone likes.

g) Everyone does not like someone.

**2. **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.

**3.** Let *B(x)*, *E(x)* and *G(x)* be the statements
"*x* is a book, " "*x* is expensive, "and
"*x* is good, " respectively. Express each of the following
statements using quantifiers; logical connectives; and *B(x)*, *E(x)*
and *G(x)*,where the universe of discourse is the set of all objects.

- No books are expensive.
- All expensive books are good.
- No books are good.
- Does (c) follow from (a) and (b)?

**4.** Let *G(x)*, *F(x)*, *Z(x)*, and *M(x)*
be the statements "*x* is a giraffw, " "*x* is 15 feet or higher,
""*x* is in this zoo, " and "*x*
belongs to me," respectively.
Suppose that the universe of discourse is the set of animals. Express each of
the following statements using quantifiers; logical connectives; and *G(x)*,
*F(x)*, *Z(x)*, and *M(x)*.

- No animals, except giraffes, are 15 feet or higher;
- There are no animals in this zoo that belong to anyone but me;
- I have no animals less than 15 feet high.
- Therefore, all animals in this zoo are giraffes.
- Does (d) follow from (a), (b), and (c) ? If not, is there a correct conclusion?