: is a book.
: is diligent.
: is expensive.
: is a student.
: likes .
(a) Not everything is a book.
(b) All students are diligent only if they like books.
(c) For a student to like a book it is necessary that the book is not expensive.
(d) Some student likes only expensive books.
(e) No student likes expensive books.
2 (a) Translate into predicate logic wffs the argument given below using the given atomic formulas.
Explain what inference rule is used in each step of your reasoning. [10]
Argument:
Every PC has a disk. Some PCs have a rewritable CD.
Therefore some PCs
have both a disk and a rewritable CD.
Atomic Formulas to be Used:
: has a disk.
: is a PC.
: has a rewritable CD.
(b) Prove that the argument is correct by deriving the conclusion
from the hypotheses step by step using one inference rule at a time. [10]
3. Prove by induction that
for all natural number . [20]
4. Recursively define the set of natural numbers { : }. [20]
5. Let be a set of functions that map the set of real numbers to the set of real numbers.
A binary relation on is defined as follows:
for any functions and in ,
if and only if
.
Prove that is an equivalence relation. [20]
Hint:
If
and
are finite,
and not equal to ,
then
=
,
and if
, then
=