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30 The Nuts and Bolts of Proof, Third Edition
This statement specifies that for Lto be the Hmit every ^ > 0 must have
a certain property. Therefore, to construct "not A," one must require that
there is at least one ^ > 0 without that property. Thus,
"Not A". The real number Lis not the limit of the function/(x) at the
point c if there exists at least one s > 0 such that for all 5 > 0 there exists
an X with 0\x — c\<8 and \f(x) - L\>£.
For more details on the definition of limits, and proofs regarding them,
see the section on Limits.
EXERCISES
Given the following statements, negate them:
L The function/is defined for all real numbers.
2. Let X and y be two numbers. There is a rational number z such that
x-\-z = y.
3. The function/has the property that for any two distinct real numbers
X and y,/(x) 7^/(3;).
4. The equation P{x) = 0 has only one solution. (Assume it is known that
the equation has at least one solution.)
5. All nonzero real numbers have nonzero opposites.
6. For every number n> 0, there is a corresponding number M„ > 0 such
that/(x) > n for all real numbers x with x > M„. (To understand this
statement better, you might want to use a graph in the Cartesian
plane.)
7. Every number satisfying the equation P{x) = Q{x) is such that |x|<5.
8. The equation P{x) = 0 has only one solution. (Check Exercise 4.)
9. The function/is continuous at the point c if for every e > 0 there is a
8>0 such that if\x-c\<8 then |/(x) -f{c)\<s.
10. For every real number x the number/(x) is rational.
Given the following statements, construct their (a) contrapositive, (b)
converse, and (c) inverses.
IL If X is an integer divisible by 6, then x is divisible by 2.
12. If a quadrilateral is not a parallelogram, then its diagonals do not
bisect.
13. If the two polynomials:
P(x) = «„x" + an_ix""^ H
h aix + ao and
are equal for all real numbers x, then ai = fcj,for all /, with 0 < i < n.