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April 30, 1998
1. Express the assertions given below as a proposition of a predicate logic using
the following predicates. The universe is the set of numbers.
N(x): x is a natural number.
P(x): x is a positive number.
(a) Not every number is positive.
(b) A number is natural only if it is positive.
(c) Some natural numbers are positive.
(d) 10 is a natural number and it is positive.
(e) Only positive numbers can be natural.
2. Prove that for sets A and B, .
3. Prove by induction that 
for all natural numbers n.
4. Recursively define the following relations:
(a) Relation " x < y" on the set of natural numbers.
(b) Relation "x is a multiple of y" on the set of natural numbers.
5. Fill in the table below with "Y" if the relation has the corresponding property, else wit
h "N".
