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I need help converting conditional statements into logical notation!

  1. Dec 5, 2007 #1
    i need to covert the following conditional statements into logical notation using propositional connectives and quantifiers:

    a) A has at most one element


    b)A is a singleton


    c)ø ∈ A

    you dont have to give me the answers, just help me get started or give me some hints
     
  2. jcsd
  3. Dec 5, 2007 #2

    cristo

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    Think of the cardinality of A.

    What is a singleton? Suppose A has two elements; what can you say about these elements?
    This says "the empty set is a member of A." This doesn't make sense, to me; don't you mean "the empty set is a subset of A?"

    you dont have to give me the answers, just help me get started or give me some hints[/QUOTE]
     
  4. Dec 5, 2007 #3
    this is what ive come up with:

    a) ∀x(x ∈ A → (x⇔ø v x ⇔ n))

    b) ∀z(z ∈ A ⇔ z = x)

    C) i didnt mistype, "ø ∈ A" is what the question said. i guess it's just a typo by the prof.

    let me know what you think of the two answers i do have though.

    thanks a lot!
     
  5. Dec 5, 2007 #4

    CRGreathouse

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    ø ∈ A is quite sensible; it's used in the canonical set-theoretic construction of Peano arithmetic, for example. But I'm not sure what you'd need to do to rewrite it.

    These have free variables, which I don't think you want. For the first one, I'd expect something like ∃n∀x (x ∈ A → x=n). Also, I'm not at all sure what you intend by "x⇔ø", which is surely not the same as your use of the double arrow in the second formula.
     
  6. Dec 6, 2007 #5

    cristo

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    Fair enough. So; what does it mean?
     
  7. Dec 6, 2007 #6

    CRGreathouse

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    "The empty set is a member of A", what else? You might use the following definitions for numbers, for example:

    0 = ø
    S(n) = n U {n}

    So that
    1 = {ø} U ø = {0}
    2 = {0} U {{0}} = {0, {0}} = {0, 1}
    3 = {0, 1} U {{0, 1}} = {0, 1, 2}
    . . .

    "ø is a subset of A" is true for all sets A, but "ø is a member of A" is true for only some A. "ø ∈ ø" is false, for example; nothing is in the empty set, not even the empty set.
     
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