Principles of Mathematics in Operations Research

132 9 Number Systems

1 !• ^ %- -%f y >y y ^

Fig. 9.2. Uncountability equivalence of (a,b) and (0,1)

-1 +1

Fig. 9.3. The correspondence between (-1,1) and '.

Example 9.6.11 / : R i-» (—f,f), /(x) = arctan(rr) is a i-i correspon-

dence, i.e. fix) is 1-1 and onto. Refer to Figure 9.4-

*=

arctan(x) ^^-^

7l<2

j, *arctan(x) x

~~rJ2

Fig. 9.4. The correspondence between (-f ,f) and

Proposition 9.6.12 // (a, b) is any open interval, then

(0,l)~(a,&)~R~[0,l)-

Proof.

3/:(0,l)^[0,l)isl-l(/(a:) = a;).

3g : [0,1) ^ R is 1-1 {g(x) = x).

1:R4 (0,1) is 1-1 and onto (f(x) = x).

[0,1) 4M4 (0,1) is 1-1.

By Cantor-Schruder-Bernstein Theorem [0,1) ~ (0,1). •