- #1
nomadreid
Gold Member
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Is the following (or, the following after any minor errors are corrected) a bijection from the unit square S=[0,1]X[0,1] to the line L=[0,1], or only an injection? If only an injection, are the excluded points in L countable?
[1] Let L be identified with the set of real numbers r, 0 ≤ r ≤ 1, whereby r is in unique decimal form 0.r1r2r3... , whereby any representation as an infinite sequence 0.s1s2...snsn+100000..., where sn≠0 & n≥ 1, is excluded, as it is identified with 0.s1s2...(sn-1)99999... (0 remains 0.000...)
[2] Let each point in S be identified with the ordered pair (a,b), with a, b∈L ,
a = 0.a1a2a3... , and
b = 0.b1b2b3... ,
[3] Then the function is (a,b) to c, with c =0.a1b1a2b2a3b3... ,
that is, if c= 0.c1c2c3... then for n≥1 , n, c2n-1=an & c2n=bn.
(or, to put another way, if a = ∑i=1∞ai×10-i & b = ∑i=1∞bi×10-i, then c = ∑i=1∞(ai×10-2i+1 + bi×10-2i)
(Corrections in the details would be welcome.)
[1] Let L be identified with the set of real numbers r, 0 ≤ r ≤ 1, whereby r is in unique decimal form 0.r1r2r3... , whereby any representation as an infinite sequence 0.s1s2...snsn+100000..., where sn≠0 & n≥ 1, is excluded, as it is identified with 0.s1s2...(sn-1)99999... (0 remains 0.000...)
[2] Let each point in S be identified with the ordered pair (a,b), with a, b∈L ,
a = 0.a1a2a3... , and
b = 0.b1b2b3... ,
[3] Then the function is (a,b) to c, with c =0.a1b1a2b2a3b3... ,
that is, if c= 0.c1c2c3... then for n≥1 , n, c2n-1=an & c2n=bn.
(or, to put another way, if a = ∑i=1∞ai×10-i & b = ∑i=1∞bi×10-i, then c = ∑i=1∞(ai×10-2i+1 + bi×10-2i)
(Corrections in the details would be welcome.)