Prove that the following sets have cardinality c.?

a) (-∞,b), for any real number b

b) the reals minus zero

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  • 1 decade ago
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    Homework.

    Steps: (some of which you may know already)

    I. Show that (0,1) is in one-to-one correspondence with R.

    II. Show that (-infty,b) is in one-to-one correspondence to (0,1).

    This finishes part a).

    III. Show that (0,1) union (1,2) is in one-to-one correspondence to (0,1).

    IV. Show that R\{0} is in one-to-one correspondence to (0,1) union (1,2).

    This will finish part b).

    ALTERNATIVELY (if you know the Cantor Bendixon theorem).

    Since (-infty,b) contains (b-1,b) it has cardinality at least that of (0,1), i.e. c. Since it is a subset of R, it has cardinality at most c. Thus it has cardinality c.

    Since R\{0} contains (0,1) it has cardinality at least c. Since it is contained in R, it has cardinality at most c. Thus it has cardinality c.

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