Principles of Mathematics in Operations Research

9.6 Problems 133

Definition 9.6.13 Roughly speaking, the cardinality of a set (or cardinal

number of a set) is the number of elements in this set.

IfX = 9, Card(X) = 0,

ffX~Jn = {l,2,...,n}, Card(X) = n,

If X ~ N (i.e. countable), Card(X) = Ko (aleph zero),

IfX~R, Card(X) = Nx (aleph one).

Definition 9.6.14 Let m and n be two cardinal numbers We say m -< n if

there are two sets X and Y 3 Card(X) = m, Card(Y) = n.

Remark 9.6.15 The list of cardinal numbers:

0-<l-<2^----<n-<----<N 0 ^:tt 1 =c.

Remark 9.6.16 Question: 3? a cardinal number between Ko and Hi?

The answer is still not known. Conjecture: The answer is no!

Question: Is there a cardinal number bigger than Hi?

The answer is yes. Consider P(R) : the set of all subsets of R (power set

of R). Hi = Card(R) -< Card(P(R)). We know if Card(X) = n, then

Card(P(X)) = 2n. Analogously Card(P(N)) = 2*° = Hx. Then, we can say

that Card(P(R)) = 2*^1 = H 2.

Problems

9.1. Let A be a non-empty subset of R which is bounded below. Define — A —

{-x : x G A}. Show that inf A = - sup(-A).

9.2. Let b y 1. Prove the following:

a) Vm,n G Z with nyQ, (bm)lln = (bl'n)m.

b) Vm,n G Z with n y 0, (bm)n = bmn = (bn)m.

c) Vn G Z with n y 0, ll'n = 1.

d) Vn,3£Z with n,qy0, b^1 /^ = (fe^1 /")^1 ^ = (blli)lln.

e) \tp,q€l bP+q = &>&.

9.3. Do the following:

a) Let m,n,p,q G Z with n -< 0, q y 0 and r = ^

(ypy/i using the above properties.

b) Prove that bT+s = brbs if r and s are rational.

c) Let x G R. Define B(x) = {bl : t G Q, t < x}

sup B(r).

d) Show that fox+y = bxW Vx, y G R.

9.4. Fix by 1 and y y 0. Show the following:
a) VnGN, bn -1 yn(b-l).
b) (b - 1) h n{bl'n - 1). Hint: Vn G N, ft^1 /" y 1 holds. So replace (b y 1)

= E. Show that (6"^1 )^1 /" =

Show that if r G Q, 6r =