- #1
talolard
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Homework Statement
What is the cardinality of the set of all continuous real valued functions [tex] [0,1] \rightarrow R [/tex].
The Attempt at a Solution
In words:
I will be using the Cantor Bernstien theorem. First the above set, let's call it A, is lesser then or equal to the set of all functions from R to R which has a cardinality of [tex] \aleph ^ \aleph [/tex]
Also, the cardinality of all continuous functions [tex] [0,1] \rightarrow R [/tex]. is lesser then or equal too the cardinality of the set of all functions, continuous and not continuous, from [tex] [0,1] \rightarrow R [/tex].
The cardinality of the segment [0,1] is [tex] \aleph [/tex] and so the cardinality of all functions [tex] [0,1] \rightarrow R [/tex]. is also [tex] \aleph ^ \aleph [/tex] and so by the cantor bernsiten theorem the cardinality of A is also [tex] \aleph ^ \aleph [/tex]
In formal math:
I use C to denote continuous functions.
[tex] \aleph ^ \aleph =|R^R| \geq |R^{C[0,1]}| = |C[0,1] \rightarrow R| = |R^{C[0,1]}| \leq |R^{[0,1]}|=\ \aleph ^ \aleph [/tex]
Is this correct? and formal anough?
Thanks
Tal