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Math Help - Re-write this sentence in logical notation

  1. #1
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    Re-write this sentence in logical notation

    "You can fool all of the people some of the time, and you can fool some of the people all of the time, but you can't fool all of the people all of the time" How do I rewrite this in logical notation? The next question is which statement is true: the original one or its negation? I don't know how to start the logical notation here..
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  2. #2
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    Quote Originally Posted by qtpipi View Post
    "You can fool all of the people some of the time, and you can fool some of the people all of the time, but you can't fool all of the people all of the time" How do I rewrite this in logical notation? The next question is which statement is true: the original one or its negation? I don't know how to start the logical notation here..
    Let P(x): x is a person; T(t): t is a time; F(x,t): You can fool x at t.

    1. You can fool all of the people some of the time.
    \forall x (\text{P(x)} \rightarrow \exists t (\text{T(t)} \wedge \text{F(x,t)}))

    2. You can fool some of the people all of the time.
    \exists x (\text{P(x)} \wedge \forall t (\text{T(t)} \rightarrow \text{F(x,t)}))

    3. You can't fool all of the people all of the time.
    \neg \forall x \forall t((\text{P(x)} \wedge \text{T(t)}) \rightarrow \text{F(x,t)})

    Finally, you need to combine 1,2 &3.
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