168 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5- Recall (Definition 5, Article 1.1) that the power set 9(A) of a set A is the
set of all subsets of A. Thus X E 9(A) if and only if X c A.
*(a) Prove that, for any sets A and B, if A E B, then 9(A) G 9(B).
(b) Prove that, for any sets A and B, 9(A n B) = 9(A) n 9(B). - (a) Use the definition of interval (Example 2) to show that each of the first
eight types of sets listed in Definition 3, Article 1.1 (e.g., [a, b], (a, GO), etc.) is an
interval. (Note: Assume throughout this proof that if p, q, and r are any real
numbers with p 5 q and q 5 r, then p I r.)
(b) Prove that if {Inln = 1,2,.. .) is a collection of intervals indexed by the set
of all positive integers, then r),"=, In is an interval.
(c) Prove or disprove: If I, and I, are intervals, then I, u I, is an interval. See
also Exercise 15, Article 5.3.) - (a) Prove that the curve C, = {(x, IxI)Ix E R} is symmetric with respect to the
y axis, but not to the x axis.
(b) Prove that the curve C2 = {(x, x3)Ix E R) is symmetric with respect to the
origin, but not to the x axis.
(c) Let f be a real-valued function with domain R. Prove that f is odd; that
is, f ( - x) = - f (x) for all x E R, if and only if the set C, = {(x, f (x)) 1 x E R) is
symmetric with respect to the origin. - A subset C of R x R is said to be symmetric with respect to the point (h, k) if
and only if, whenever (x + h, y + k) E C, then (-x + h, - y + k) E C.
(a) Prove that if C is the graph of a function y = f(x), then C is symmetric with
respect to the point (h, k) if and only if f(-x + h) = 2k - f(x + h) for every x
such that x + h is in the domain off.
*(b) Show that the graph of the function y = f(x) = 1 + (l/(x - 1)) is symmetric
with respect to the point (1, 1).
(c) Use Exercise 18, Article 5.1, to conclude that the graphs of the functions cos- ',
tan-', and cot-' are each symmetric with respect to the point (0,71/2).
' 6. A function f, mapping real numbers to real numbers, is said to be one to one on
an interval I if and only if, for any real numbers x, and x, in the interval I, if
f(x,) = f (x,), then x, = x,.
(a) Prove that f(x) = x2 is not one to one on R.
(b) Prove that, if M # 0, then the linear function y = f(x) = Mx + B is one
to one on R.
(c) Prove that if f is increasing on an interval I, then f is one to one on I.
(Hint: contrapositive)- Consider the curve C, in the xy plane described parametrically by the equations
x = cos t, y = sin t, where t is any real number; that is, C, = {(cos t, sin t)l t E R).
Note that a point (x, y) in the xy plane is on C, if and only if there exists a real
number t such that x = cos t and y = sin t.
(a) Let C, be the curve ((x, y) E R x Rlx2 + y2 = 1). Use well-known properties
of sine and cosine to prove that C, c C,. (In fact, these curves are the same;
that is, C, = C,. We will consider the reverse inclusion in Article 6.1.)