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Math Help - English to Proposition Statements

  1. #1
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    English to Proposition Statements

    I am given a statement that says "All teachers wear glasses." And to put it in logical statement, I had a few lines that I thought they sound the same:

    Let all humans be in the set of H.
    Let all teachers be in the set of T.
    Let "is a teacher" be T(x).
    Let "does wear glasses" G(x).

    <br />
\forall \; x \epsilon H,\; T(x) \to G(x)<br />
    <br />
\forall \; x \epsilon T,\; G(x) \to T(x)<br />
    <br />
\forall \;x \epsilon T,\; G (x)\leftrightarrow T(x)<br />
    <br />
\forall \;x \epsilon H,  x \epsilon T, \; G(x)<br />
    <br />
\forall \;x \epsilon T,\; G(x)<br />
    <br />
\forall \;x \epsilon H,\: x \epsilon T,\; G(x) \wedge  T(x)<br />

    Out of these statements, which ones are correct? All of these sound logical to me somehow. But I am quite sceptical over the first 2 statements because I remember that p->q is not equals to q->p albeit I have a different set of domain.

    thanks...
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  2. #2
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    The first one looks correct.
    The second one is not correct. For example, let Bob be a teacher that doesn't wear glasses. Then Bob is in T, G(Bob) is false, and T(Bob) is True.
    The third is correct, but it's overcomplicated.
    The fourth and sixth don't make sense to me since you're quantifying over the same variable twice.
    The fifth is correct.
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  3. #3
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    Actually, I am always very confused with the "-->" logic. Does it mean like "if .... then"? So wouldn't it be "IF 'wear glasses' then 'is a teacher'" otherwise, not teacher?

    Also, for the forth and sixth one, am I declaring the quantifier wrongly? I was trying to have like "For all x in H, that are also in T".
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  4. #4
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    Quote Originally Posted by xEnOn View Post
    Actually, I am always very confused with the "-->" logic. Does it mean like "if .... then"? So wouldn't it be "IF 'wear glasses' then 'is a teacher'" otherwise, not teacher?

    Also, for the forth and sixth one, am I declaring the quantifier wrongly? I was trying to have like "For all x in H, that are also in T".
    Yes if then.
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  5. #5
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    Quote Originally Posted by xEnOn View Post
    Actually, I am always very confused with the "-->" logic. Does it mean like "if .... then"? So wouldn't it be "IF 'wear glasses' then 'is a teacher'" otherwise, not teacher?

    Also, for the forth and sixth one, am I declaring the quantifier wrongly? I was trying to have like "For all x in H, that are also in T".
    The conditional statement p\rightarrow q is read "implies" or "if...then..." and is only false if p is true and q is false.

    The following statement doesn't make sense:
    \forall \;x \epsilon H, x \epsilon T, \; G(x)

    Maybe you mean \forall x(x \epsilon H \rightarrow (x \epsilon T\wedge G(x)))?

    In any case, this doesn't correspond to what you're trying to say.
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  6. #6
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    So wouldn't it be "IF 'wear glasses' then 'is a teacher'" otherwise, not teacher?
    I am not exactly sure what you mean, but "All teachers wear glasses" is equivalent to "For every person x, if x is a teacher, then x wears glasses."

    Also, for the forth and sixth one, am I declaring the quantifier wrongly? I was trying to have like "For all x in H, that are also in T".
    \forall \;x \epsilon H, x \epsilon T, \; G(x) is wrong because it is not a syntactically well-formed formula (wff). If you have some wffs P and Q, then one can build the following wffs: P\land Q, P\to Q, \forall x\,P and so on. Also, \forall x\in H,\,P is (most likely) a contraction for \forall x\,(x\in H\to P). Your formula does not satisfy these construction rules.

    I was trying to have like "For all x in H, that are also in T".
    Then formula #4 should be \forall x\, (x\in H\to (x\in T\to G(x))), or, using the contraction, \forall x\in H\, (x\in T\to G(x)). Another equivalent form is \forall x\, (x\in H\land x\in T\to G(x)). These three formulas do say that all teachers wear glasses.
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