176 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5- (a) Prove that if a subset C of R x R is symmetric with respect to both the x
axis and the y axis, then it is symmetric with respect to the origin.
(b) Prove that if a subset C of R x R is symmetric with respect to both the origin
and the y axis, then it is symmetric with respect to the x axis. - Assume it is known that cos (x - y) = cos x cos y + sin x sin y for all real num-
bers x and y. Use the additional facts that cos2 x + sin2 x = 1 for all x E R,
cos (42) = 0, sin (42) = 1, cos 0 = 1, sin 0 = 0, and sin (-42) = - 1, to prove:
(a) cos ((42) - y) = sin y for all y E R (This is the result we assumed in
Example 4.)
(b) cos (-y) = cqs y for all y E R (i.e., cosine is an even function.)
(c) cos (X + (42)) = - sin x for all x E R
(d) sin (-x) = -sin x for all x E R [i.e., sine is an odd function. Hint: Use
the results of (a) and (c).]
(e) sin (x + (42)) = cos x for all x E R [Hint: Use (a).]
(f) cos (x + y) = cos x cos y - sin x sin y for all x, y E R
(g) sin (x + y) = sin x cos y + cos x sin y for all x, y E R [Hint: Use (a) and
Example 4.1
(h) sin (x - y) = sin x cos y - cos x sin y for all x, y E R
(i) sin 2x =^2 sin x cos x for all x E R - Use the results from Exercise 9 to prove further that:
(a) sin x - sin y = 2 cos ((x + y)/2) sin ((x - y)/2) for all x, y E R
(b) cos 2x = 1 - 2 sin2 x = 2 cos2 x - 1 for a11 x E R
(c) cos2 x = (1 + cos 2x)/2 for all x E R
(d) sin2 x = (1 - cos 2x)/2 for all x E R
(e) tan (x - y) = (tan x - tan y)/(l + tan x tan y), whenever x, y E R and x #
y + [(2n + l)?c/2], for any integer n
11. For given real numbers x and y we define:
ysx Y, y5x
max(x,y) = xvy = and min (x, y) = x A y =
x, xly
Prove that, for any real numbers x, y, and Z:
(a) XA(YAZ) =(XAY)AZ
(b) (xAy)+(xvy)=x+Y
(c) (-x)A(-Y) = -(XVY)
(d) (xvy)+z=(x+z)v(y+z)
(e) If z > 0, then z(x v y) = (zx) v (zy)
12. (a) Complete the proof from Example 5 that lxyl = 1x1 lyl for all real numbers x
and y.
(b) Recall from Exercise 8, Article 5.1, that (x + yl I 1x1 + Iyl for all x, y E R and
1x1 9 y if and only if -y I x 5 y, where x, y E R and y 2 0. Prove that:
(i) IxI=xv(-x) foranyx~R
(jj) xvy=~(x+y+~x-yl) foranyx,y~R
(iii) x A y = f (x + y - Ix - yl) for any x, y E R
(iv) (x v y) - (x A y) = lx - yl for any x, y E R