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Problem 2 Of Set Theory

1. Consider the saying: \You can please some of the people all of the time
and all of the people some of the time, but not all of the people all of the
time."

(a) Express this sentence using quantifiers and predicates . Be sure to
define your notation.

(b) Negate the expression derived in (1a).

2. Let S be a finite set. Show that the following two definitions of submodu-
larity are equivalent . A function f : 2s
R is submodular if,

3. Prove that the rational numbers and the integers have the same cardinal-
ity.1
4. ( Fibonacci Numbers . ) The definition of Fibonacci numbers can be found
in page 5 of the reading note #3. Let
Prove using induction:

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