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Math Help - Help with a proof

  1. #1
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    Help with a proof

    Hi,

    I need some help with a homework problem involving a proof
    i think it is pretty simple but I've never done a proof before so
    I'm at a loss as to where to even begin with this...

    I have to prove the following:



    Any help greatly appreciated,
    regards

    -dc
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by dogcow View Post
    Hi,

    I need some help with a homework problem involving a proof
    i think it is pretty simple but I've never done a proof before so
    I'm at a loss as to where to even begin with this...

    I have to prove the following:



    Any help greatly appreciated,
    regards

    -dc
    i'm not sure i fully understand the notation you are using. but by the definition of the union, x \in (P(x) \cup Q(x)) \Longleftrightarrow \left[  (x \in P(x)) \cup ( x \in Q(x))\right]
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  3. #3
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    Quote Originally Posted by Jhevon View Post
    i'm not sure i fully understand the notation you are using. but by the definition of the union, x \in (P(x) \cup Q(x)) \Longleftrightarrow \left[  (x \in P(x)) \cup ( x \in Q(x))\right]

    i didnt mean the "V" to be taken as a union its an disjunction (or)
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  4. #4
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    To prove \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right] \equiv \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right] do side at a time.
    If \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right] is true then for some t \left[ {P(t) \vee Q(t)} \right] if P(t)\quad  \Rightarrow \left( {\exists x} \right)\left[ {P(x)} \right]
    or if Q(t)\quad  \Rightarrow \left( {\exists x} \right)\left[ {Q(x)} \right].
    Thus we have \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right].

    Say that \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right] is true.
    Then for some s we have P(s) or for some u we have Q(u).
    In the first instance: P(s)\quad  \Rightarrow \quad P(s) \vee Q(s) \Rightarrow \quad \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]\quad .
    In the second, Q(u)\quad  \Rightarrow \quad Q(u) \vee P(u) \Rightarrow \quad \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]\quad
    So in either we get \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]
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  5. #5
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    Quote Originally Posted by Plato View Post
    To prove \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right] \equiv \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right] do side at a time.
    If \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right] is true then for some t \left[ {P(t) \vee Q(t)} \right] if P(t)\quad  \Rightarrow \left( {\exists x} \right)\left[ {P(x)} \right]
    or if Q(t)\quad  \Rightarrow \left( {\exists x} \right)\left[ {Q(x)} \right].
    Thus we have \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right].

    Say that \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right] is true.
    Then for some s we have P(s) or for some u we have Q(u).
    In the first instance: P(s)\quad  \Rightarrow \quad P(s) \vee Q(s) \Rightarrow \quad \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]\quad .
    In the second, Q(u)\quad  \Rightarrow \quad Q(u) \vee P(u) \Rightarrow \quad \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]\quad
    So in either we get \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]

    is this the right way to prove something or do i have to use actual examples? I mean I dont get how this proves it beyond just restating it in a different way. Or are you saying I have to prove each side?
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  6. #6
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    Quote Originally Posted by dogcow View Post
    is this the right way to prove something or do i have to use actual examples?
    Your instructor and or the textbook must answer that question.

    Quote Originally Posted by dogcow View Post
    I mean I dont get how this proves it beyond just restating it in a different way. Or are you saying I have to prove each side?
    I taught formal logic for many years. This is an outline of what complete proof might look like. I did leave out some justifying citations because no two authors agree on what some operations are called. You must go to your textbook for details.

    But I must be honest and tell you that it seems that you are not quite sure of what it means to prove a logical proposition.
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  7. #7
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    Quote Originally Posted by Plato View Post
    Your instructor and or the textbook must answer that question.

    I taught formal logic for many years. This is an outline of what complete proof might look like. I did leave out some justifying citations because no two authors agree on what some operations are called. You must go to your textbook for details.

    But I must be honest and tell you that it seems that you are not quite sure of what it means to prove a logical proposition.
    Yes as I mentioned in my first post I've never done that before...which is why i was asking for some help, because I dont understand how to do it. Can you give me a brief explanation my text is no help
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