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    Denumerability and induction

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    1. Show that if A and > are denumerable disjoint sets then A u > is denumerable

    2. Show that every set of cardinalty c contains a denumerable subset

    3. Show by induction that 6 divides n^3 - n for all n in N

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    https://brainmass.com/math/basic-algebra/denumerability-induction-61007

    Solution Preview

    1. Show that if A and > are denumerable disjoint sets then A u > is denumerable
    Proof:
    For better notation, I replace > with B. So the statement is: if A and B are denumerable disjoint sets, then A U B is denumerable.
    We note, a set is denumerable if and only if it is finite or coutable infinite. So we have 4 cases.
    Case 1: Both A and B are finite. Let |A|=n and |B|=m, since A and B are disjoint, then |A U B|=m+n is also finite. So A U B is denumerable.
    Case 2: A is finite, B is countable infinite. Let |A|=n, ...

    Solution Summary

    There are three proofs here: one regarding denumerability, one regarding cardinality and denumerable subsets, and a proof of divisibility by induction.

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