epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

68 The Nuts and Bolts of Proof, Third Edition
where ^>0 and 0<r <b, and q' >0 and 0</ < fe.
Without loss of generahty, we can assume that / > r.
Thus,
b{q -q')=:r'
-r>0.
Because r' > r' — r, we have b> r' — r.
Therefore,
Dividing by b we obtain
b>b(q- q') > 0.
1> q-q' >0.
This impHes that q-q^ = 0, or q = q\
lfq-q^ = 0, then r' - r == 0. So, / = r.
The theorem is now completely proved.
EQUALITY OF SETS
A set is a well-defined collection of objects. The objects that belong to a set
are called the elements of the set. If x is an element of a set A, we write x e A.
The empty set is a set with no elements, usually represented by either
0 or {}.
Sets can be described in several ways. We can either provide a Hst of the
elements (roster method) or we can Hst the property (properties) the elements
must have in order to belong to the set (constructive method). If we are
listing the elements of a set, the order of the Hsting is irrelevant, and the same
element should appear only once (repeated elements do not count as distinct
elements).
The roster method is not very practical when the set has a large number
of elements, and it is impossible to use when the set has an infinite number of
elements. In this latter case, we should Hst enough elements for a pattern to
emerge and then use "..."; for example, it would be a bad idea to write:
^-{3,5,7,...}
because we do not have enough information to decide whether A is the set of
odd numbers larger than 1, or the set of prime numbers larger than 2.
When we use the constructive method, usuaHy we have two parts in
the description. The first part specifies which kind of objects we are