1549901369-Elements_of_Real_Analysis__Denlinger_

606 Appendix A • Logic and Proofs

PART B: Write the negation of each proposition in Part A, first in sym-
bols, and then in words. Give a complete analysis in each case.

PART C:

l. Let I denote the set of integers, E denote the set of even integers, and

0 denote the set of odd integers. Translate each of the following into

words, and tell whether it is true or false. Then state the negation, first

in symbols and then in words.

(a) '<Ix EI, 3y E 0 3 x + y EE.

(b) '<Ix E 0 , 3y E 0 3 x+y EE.

(c) 3 yE13'VxE13x+y=x.

(d) '<Ix EI, 3 y EI 3 x + y = 0.

2. For each of the following, tell whether the statement is true or false, and
   then state t he negation.

(a ) For each positive real number x, there is a positive real number y such

that y is less than x.

(b) For each positive integer x, t here is a positive integer y such that y is

less than x.

( c) There is a real number x such that for each real number y , x + y = 0.

A.3 Strategies of Proving Theorems

There is , of course, no such thing as a general method that works in proving all
theorems. Proving theorems is a creative art that defies all attempts to reduce
it to a routine procedure. Nevertheless, it is possible to make some helpful
suggestions and describe some useful strategies. That is about all we attempt
in this section.
First, a theorem usually has hypotheses and always has a conclusion. The
theorem claims t hat t he hypotheses provide sufficient evidence to guarantee
the truth of the conclusion. Schematically, a theorem takes the following form:

Hypothesis #1

Hypothesis #2

Hypothesis #n

:. Conclusion

or, symbolically,

Hn

... c