Question

10. (20 points) Show that the sets (0,1) and (1,0) have equal cardinality by describing a bijection from one to the other. Describe your bijection with a formula (not as a table).

Intro to proofs, thanks!

Answer 1

first we that open When Here 우 4 Here Given that the sets [0, 1 and (1, 2) have Squar cosidinolity by desceribing a bijection Here and prove closed poterials have same costdinality Define o map f: [0, 1] [0, 1] by PO) = (x+12/3 Then map is strictly incoreasing bence injective When the imago of f is contained in the integral. theore [53e corthso the maps for:[0,1] + [0,1] and fo: [0,1] [0,1] obtained from f by restricting the Cochomair both injective. Since the inclusions into also injective. Thusa o bijedtion can be mode. such that ho :[0, 1] [0, 1] and ho :[0, 1] → [0, 1] finally here the map st: (0,1) - [0,1) defined by 9x) = 2 x bijection Thans 1o, 1) (0, 1), [0, 1], (0,1),(0, 1), (0.1) are samo coidinalily Now, we show that cou) and (1,0) have Squal Cardinality Since Card Cord 4 (6) .. Hence proved earlier Howl, Define map f:(0, 0 + (1,00) by f(x) = 7 ace [0,1] are is a a .csSearned-witireamscanner

Here f(a)= Yat flot) - 2 07 fe) 1 11 = 1 CL since the foc is monotonically. decoreasing every Element Encodemain (1700) have pre- Image in domain Co, 12 fice, 1) = (1,0), f(x) Yox is bijection (o, and (1,6) have Equal Cordinality * [04] and (1,00) have Equal Cardinality. Thus - CScanned with CamScanner