My Math Forum Unit Interval vs. Unit Square Cardinality Question

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 May 13th, 2011, 04:28 PM #1 Newbie   Joined: May 2011 Posts: 1 Thanks: 0 Unit Interval vs. Unit Square Cardinality Question Hello, I'm currently studying sets and cardinality on my own, and I've come across some non-intuitive results I can't comfortably "fix". I've come to understand that the unit interval [0, 1] and the unit square [0, 1] x [0, 1] have the same cardinality because a one-to-one correspondence can be created between them. The only example I have seen of such a correspondence is as follows: Any real number, A, between 0 and 1 inclusive can be represented as digits: 0.a1a2a3a4a5... If (x, y) is a point in the unit square written as (0.x1x2x3x4... , 0.y1y2y3y4y5...) the corresponding point on the unit interval can be written as 0.x1y1x2y2x3y3..., basically "interlocking" the coordinates together. In reverse, A is mapped to (0.a1a3a5..., 0.a2a4a6...) on the square. My problem comes from the ability of some rational numbers to have two decimal expressions. For example, suppose I want to find the point in the unit square where four-fifths is mapped to using the above method. Four-fifths can be written as 0.8000.... or 0.7999.... Using the first expression, A = 0.8000... and x = 0.800... and y = 0.000..., so 0.8 maps to (0.8, 0) Using the second expression, A = 0.7999... and x = 0.7999... and y = 0.999..., so 0.7999... maps to (0.7999..., 0.999...) or (0.8, 1) Two different map results for the same input (just expressed in a different way)? How is this possible? You can go on to see more surprising results in the reverse direction. Mapping (0.8, 0.2) to the unit interval, the coordinate pair can take on four different expressions resulting in four unique mapping locations: (0.8, 0.2) maps to (0.82) (0.7999..., 0.2) maps to (0.72999...) (0.8, 0.1999...) maps to (0.81099...) (0.7999..., 0.1999...) maps to (0.71999...) Does this show that the function is not one-to-one? Does this show that the two expressions one can use for the same number are actually different? How can this be justified? Thank you for your time.
 May 13th, 2011, 05:00 PM #2 Senior Member   Joined: Jun 2010 Posts: 618 Thanks: 0 Re: Unit Interval vs. Unit Square Cardinality Question miss_direction, The function is not well-defined. To remedy this, you can simply settle on one form of the decimal expansion in the ambiguous cases. Then you will recover a well-defined function which is also a bijection. -Ormkärr-
 May 13th, 2011, 09:36 PM #3 Member   Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0 Re: Unit Interval vs. Unit Square Cardinality Question If 4/5 gets mapped to two distinct points then the function is not injective.

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