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Math Help - Exercises from "Introduction to set theory" by Hrbacek, Jech 3rd ed.

  1. #1
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    Exercises from "Introduction to set theory" by Hrbacek, Jech 3rd ed.

    Hello folks,

    I'm attempting some self-study in set theory using the text mentioned above. The exercises here are quite different from those in previous texts which I've used, so I was hoping I could present some of my attempts (so far, only from the first problem set) and receive some feedback as to whether it seems I am performing them correctly. First I'll briefly state the axioms they use:


    Axiom of existence:
    There exists a set which has no elements.

    Axiom of extensionality:
    If every element of X is an element of Y and every element of Y is an element of X, then X = Y.

    Axiom Schema of Comprehension:
    Let P(x) be a property of x. For any set A, there is a set B such that x ∈ B iff x ∈ A and P(x).

    Axiom of pair:
    For any A and B, there is a set C such that x is an element of C iff x = A or x = B.

    Axiom of union:
    For any set S, there exists a set U such that x ∈ U iff x ∈ A for some A ∈ S.

    Axiom of Power set:
    For any set S, there exists a set P such that X ∈ P iff X ⊆ S.



    3.1: Show that the set of all x such that x ∈ A and x B exists.
    3.1 solution: Let P(x, A, B) be the property x ∈ A and x B. Then W = {x ∈ A | P(x, A, B)} exists by the axiom schema of comprehension.


    3.4 Let A and B be sets. Show there exists a unique set C such that x ∈ C iff either (x ∈ A and x B) or (x ∈ B and x A).
    3.4 solution: By the axiom of pair, there is some set D such that x ∈ D iff x = A or x = B. Let P(x, A, B) be the property (x ∈ A and x B) or (x ∈ B and x A). Then W = {x
    ∈ D | P(x, A, B)} exists by the axiom schema of comprehension. If K is another set which satisfies x ∈ K iff either (x ∈ A and x B) or (x ∈ B and x A)
    , then x ∈ K iff x ∈ C, and so by extensionality K = C.

    3.5 a)Given A, B, and C, there is a set P such that x ∈ P iff x = A or x = B or x = C.
    b) generalize to four elements
    3.5 a) solution: Using A and B along with the axiom of pair provides us with a new set D such that x ∈ D iff x = A or x = B. Now use the axiom of pair again, this time with C alone to obtain the set E such that x ∈ E iff x = C. Now use the axiom of pair one last time, with D and E, to create the set F such that x ∈ F iff x = D or x = E. Finally, apply the axiom of union to F, which yields the set P, where x ∈ P iff x ∈ H for some H ∈ F. Thus, P = {x | x ∈ D or x ∈ E} = {x | x = A or x = B or x = C}.
    3.5 b) I'll wait to do this one until someone can verify part a).
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  2. #2
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    Re: Exercises from "Introduction to set theory" by Hrbacek, Jech 3rd ed.

    It all looks good, the only remark I have is that in 3.5a) it is unnecessary to use the axiom of pair on C because by the axiom of extensionality C=E. Note that what you have is correct, just not necessary.

    On a related I am also working through Jech's book on set theory and it is awesome!
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