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Math Help - A couple induction set proofs

  1. #1
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    A couple induction set proofs

    Prove the following statements

    1) Prove: There is just one minimal induction set.
    I know that there cannot be two minimal induction sets, but this says there does not exit zero minimal induction sets. I'm pretty sure that I have to use the definitions of propers subsets and subsets in there somewhere, I'm just not sure where it all fits in.
    2) Prove: If C' is a subset of C, and C' is an induction set, then C' = C.
    I haven't really gotten started on this one, but I know that an induction set is a set A such that 1 is an element of A and if x is in A, then x+1 is an element of A.
    Thanks for any help you are able to provide
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  2. #2
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    Quote Originally Posted by noles2188 View Post
    Prove the following statements
    1) Prove: There is just one minimal induction set.
    Let C be the set generated by a basis B={0} and a successor function S(n) = n+1. C can be constructed as follows:
    1. Starting with B, recursively include an element using the successor function such that C_1=\{0\}, C_2=\{0,1\},,,, (You might use an empty set notation for an inductive set).
    2. Define C as C=\bigcup_nC_n.

    We shall show that an intersection of all the inductive subsets of a universe U is C. That means, C is the minimal inductive set.

    Let N= \bigcap\{\text{A : A is inductive subset of U}\}.

    To show N \subseteq C, we only need to show C is an inductive set. C includes B and we see that a S(c) is in C for every c \in C.
    Conversely, N contains B and is closed under S. Thus C \subseteq N.

    2) Prove: If C' is a subset of C, and C' is an induction set, then C' = C.
    Both C and C' are inductive sets. Since C \subseteq N (N is the smallest inductive set) and C' is an inductive set, C \subseteq C'. We also know that C' is a subset of C. Thus, C' = C.
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