200 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6(b) An integer n is said to be even if and only if there exists an integer m (nec-
essarily unique) such that n = 2m, and odd if and only if n = 2m + 1 for some
integer m (where m again is uniquely determined by n). Assume that every integer
is either even or odd, and not both even and odd.
*(i) What tautology of the propositional calculus would enable us to con-
clude, from the preceding assumption, that an integer m is even if and
only if it is not odd? Formulate and verify, by using a truth table, such
a tautology.
(ii) Prove that if n is even, then n +^1 is odd, n + 2 is even, and n +^3 is odd.
(iii) Prove that if m is even, then m2 is even.
(iv) Prove that if m is odd, then m2 is odd.- Recall from Definition 7, ff., Article 1.2, that an object x is in the cartesian prod-
uct A x B of two sets A and B, if and only if there exist elements a E A and b E B
such that x = (a, b). Prove that, if W, X, Y, and Z are arbitrary sets, then:
*(a) (X u Y) x Z = (X x 2) v (Y x Z)
(b) (Xn Y) x Z =(X x 2) n (Y x 2)
(c) (X- Y)xz qx xZ)-(YXZ)
(d) if W z X and Y G 2, then W x Y c X x 2.
[Note: The reverse inclusion in (c) is true as well. See Exercise l(e), Article 6.2.1
- [Continuation of Exercise 7(a), Article 5.21 Let C, = {(cos t, sin t) 1 t E R) and
C, = ((x, y) E R x Rlx2 + y2 = 1). Recall that (x, y) E C, if and only if 3t E R such
that x = cos t and y = sin t.
(a) Prove that C2 E C1. [Hint: Given (x, y) E C,, let t = sin-' y. Clearly y =
sin t. Use identities from Exercise 18, Article 5.1, to prove that x = cos t.]
(6) Prove that C, is symmetric with respect to the x-axis and the origin. Use
these results and Example 1, Article 5.3, to conclude that C, is symmetric with
respect to the y axis. - (Continuation of Exercise 9, Article 5.2) (a) Suppose x, y, and z are real numbers
with x < z < y. Prove that there exists a real number t, 0 < t < 1, such that z =
tx + (1 - t)y. (Hint: On what quantities would you expect the desired t to
depend?)
*(b) Prove that if S is a convex subset of R, then S is an interval. (Note: Combining
this result with Exercise 9, Article 5.2, we conclude that a subset S of R is convex
if and only if it is an interval.) - (a) Assume that the inverse of an n x n matrix, if it exists, is unique [to be
proved in Article 6.3, Exercise 5(a)]. Prove that if A, B, and C are invertible
n x n matrices, then:
(i) A- ' is invertible.
(ii) ABC is invertible.
(iii) The matrix cA is invertible, where c is a nonzero real number.
(iv) If AD = AF, where D and F are arbitrary n x n matrices, possibly non-
invertible, then D = F. (Hint: Write a proof by transitivity.)
(b) Prove that if A is any square matrix, then A can be expressed in the form
A = B + C, where B is a symmetric matrix and Cis antisymmetric [recall Exercise
1 7(f), Article 5.11.