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

Solutions for the Exercises at the End of the Sections and the Review Exercises 151
Case 2. n>Ms = ((1 + £)/s). Because ^ < jf^, we have |^ - 0| =
Moreover, (1 + Is) < (1+ 2s + s^) = (1 + sf.
Therefore, |ri±i - 0| < ^ (1 + 2£)< ^ ( 1 + 2sf = s.
SOLUTIONS FOR THE REVIEW EXERCISES
1. We are assuming that the two points, P and Q are distinct. Therefore,
the values of their x-coordinates or the values of their y-coordinates
are different; that is, either xi / X2 or yi ^ y^. We will assume that
xi 7^ X2 (geometrically, this means that the points are not on the same
vertical Hne). Because this implies that xi - X2 ^ 0, we know that
(xi - X2)^>0. The quantity (j^i - yjf is always nonnegative (because
it is the second power of a real number). Therefore,
(xi -X2)^+(3^i —yif'^. (xi -X2)^>0. This implies that d itself is a
positive number. We have proved that the given statement is true
under the assumption that xi 7^ X2. Similarly, we can prove that the
statement is true under the assumption y\ ^ yi (geometrically, this
means that the points are not on the same horizontal line). The part of
the proof for the y-coordinates is not a "must" if one observes that the
formula used to evaluate the distance is symmetric with respect to the
X- and y-coordinates (that means that we could switch the two
coordinates and the formula would not change); therefore whatever
was proved for one of the coordinates is true for the other one as well.
There is a third part of the proof that is not needed, but we want to
mention it for completeness sake. It is possible to assume that xi i=- X2
and yi ^ yi at the same time. The given statement is still true in this
case, because we have proved that it holds true when only one pair of
coordinates has different values, which is a much weaker assumption
than this last one.
2. Let a be any real number and d be an opposite of a. Then
Let h be another number such that
a + a' = a' + a = 0 (1)
^ + ft=:fc + a = 0 (2)
(Because we have two distinct numbers acting similarly on the number
a, we should wonder how they interact with each other. The answer to
this question is not evident because we only know Equations (1) and
(2), and both sets of equalities involve a. Then let us try to construct