To find the base -3 expansion of 1 2 To find a one-to-one correspondence between the Cantor set C and the interval [ 0 , 1 ] To show Cantor set is not an “all endpoints” by explicitly identifying a point in C that is not an endpoint. Concept Introduction: The Cantor set is created by starting with closed interval S 0 = [ 0 , 1 ] and removing its open middle third, i.e. interval ( 1 3 , 2 3 ) and the endpoints are kept as they are. This procedure creates the pair of the closed interval S 1 , then removing the open middle third of the intervals S 0 and S 1 creates S 2 and so on. The limiting set C = S ∞ is the Cantor set. The fractal properties of the Cantor set are C has structure at arbitrarily small scales. C is self-similar The dimension of C is not an integer C has measure zero C consists of unaccountably many points

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Chapter 11.2, Problem 5E

Interpretation Introduction

Interpretation:

To find the base ^{-3 expansion of 12}

To find a one-to-one correspondence between the Cantor set C and the interval [0,1]

To show Cantor set is not an “all endpoints” by explicitly identifying a point in C that is not an endpoint.

Concept Introduction:

The Cantor set is created by starting with closed interval S0=[0,1] and removing its open middle third, i.e. interval (13,23) and the endpoints are kept as they are. This procedure creates the pair of the closed interval S1, then removing the open middle third of the intervals S0 and S1 creates S2 and so on. The limiting set C = S∞ is the Cantor set.

The fractal properties of the Cantor set are

C has structure at arbitrarily small scales.

C is self-similar

The dimension of C is not an integer

C has measure zero

C consists of unaccountably many points

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