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Sets and Quantifiers

  1. Jan 28, 2017 #1
    1. The problem statement, all variables and given/known data
    B ∈ {P (A) | A ∈ F}. where P(A) is the power set of A and F is Family

    2. Relevant equations
    N/A

    3. The attempt at a solution
    My interpretation:
    A: an element of the Family of sets. Hence, A is a set.
    P(A): the set of all the possible unique subsets of A.
    B: an element of the set of subsets of A, P(A). Thus, B⊆A.
    ∀x[x∈B → x∈A]

    correct interpretation: ∃A ∈ F ∀x(x ∈ B ↔ ∀y(y ∈ x → y ∈ A))
    i do not understand how that came to be and why my interpretation is wrong.
     
  2. jcsd
  3. Jan 28, 2017 #2

    Math_QED

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    What is the exact problem? You didn't specify what you are having problems with.
     
  4. Jan 28, 2017 #3
    i do not understand why my interpretation is wrong. how should i read it? i've specified my problem in the attempt at a solution section of my post
     
  5. Jan 28, 2017 #4

    Stephen Tashi

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    So. what is the problem statement?

    Is it:
    ?
     
  6. Jan 29, 2017 #5
    yes. that is where i am having trouble with. the steps in the book are as follows:
    1) ∃A ∈ F(B = P (A))
    2) ∀x(x ∈ B ↔ x ⊆ A)
    3) ∃A ∈ F ∀x(x ∈ B ↔ ∀y(y ∈ x → y ∈ A))

    I don't even understand how B=P(A). I thought B ∈ P(A) such that A ∈ F. My whole interpretation of B ∈ {P (A) | A ∈ F} is B is an element of the set of all subsets of A where A is an element of F. Thus, B is a proper subset of A and not simply any subset.
    Now where in my interpretation did I went wrong? thank you.
     
  7. Jan 29, 2017 #6

    Stephen Tashi

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    You should quote the entire statement of the problem, exactly.

    You've left your advisers guessing at what the "ground rules" are for solving it.

    My guess is that this is supposed to be a symbolic representation of a statement logically equivalent to:
    "The set B is an element of the power set A and A is a element of the family of sets F"

    For those words to be a statement (be it a true or false statement) the things that are declared to have a relation must exist or else there must be some convention about how notation indicating a relation is interpreted if one or both of the things happens not to exist.


    The meaning of that notation is unclear. It might mean "There exists an ##A## such that ##A \in F## and we define ##B## to be the power set of ##A##". That notation didn't bother to say that ##F## exists. Perhaps the ground rules are that ##F## is always assumed to exist. Apparently another assumption is that if set ##A## exists then set ##P(A)## exists.

    That could be justified by the definition of ##B## as being ##P(A)##


    If we recall ##B = P(A)## then that is a correct statement, but not knowing what the exact goal of the problem is, we can't say why that is "the answer".

    Not knowing the ground rules for the problem, I can't say why your interpretation is not the answer. As far as I can see your interpretation is a correct statement if we take for granted that certain things exist. However, I don't know what things we are allowed to assume exist and I don't know the exact conditions that "the answer" is required to satisfy.
     
    Last edited: Jan 29, 2017
  8. Jan 30, 2017 #7
    this was the ground rule: "Analyze the logical forms of the following statements." besides that we should know that P(A) is the power set of A and F means family set. It no longer elaborated what other sets were in F. All we should know is A is an element of F and B ∈ {P (A) | A ∈ F}.
     
  9. Jan 30, 2017 #8

    Stephen Tashi

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    If the exact statement of the problem is "Analyze the logical forms of the following statements" then we need to know how your text materials define a "logical form". Is there a definition for "logical form"? Are there some general instructions like "When we analyze the logical form of a statement we do the following..." ?
     
  10. Jan 30, 2017 #9
    it's using quantifiers in conditional statements, or statements, and statements, combining them all, etc...
     
  11. Jan 30, 2017 #10

    Stephen Tashi

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    If the goal is to use as many quantifiers as possible then the "correct answer" beat yours in that respect (!).
     
  12. Jan 30, 2017 #11
    then I really need help to know where i went wrong lol. thanks
     
  13. Jan 30, 2017 #12
    i think i got it now. please do correct me if i misunderstood anything.

    let x be an element of B,
    Since x is an element of B and B is a subset of A then x must also be an element of A.
    Since x is an element of A and by definition of subsets, x is also a subset of A which makes it an element of P(A).
    Therefore, we can write ∀x(x ∈ B ↔ x ⊆ A).
     
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