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Thread: 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, $\displaystyle x \in (P(x) \cup Q(x)) \Longleftrightarrow \left[ (x \in P(x)) \cup ( x \in Q(x))\right]$
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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, $\displaystyle x \in (P(x) \cup Q(x)) \Longleftrightarrow \left[ (x \in P(x)) \cup ( x \in Q(x))\right]$

    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 $\displaystyle \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right] \equiv \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right]$ do side at a time.
    If $\displaystyle \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]$ is true then for some t $\displaystyle \left[ {P(t) \vee Q(t)} \right]$ if $\displaystyle P(t)\quad \Rightarrow \left( {\exists x} \right)\left[ {P(x)} \right]$
    or if $\displaystyle Q(t)\quad \Rightarrow \left( {\exists x} \right)\left[ {Q(x)} \right]$.
    Thus we have $\displaystyle \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right]$.

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

    Say that $\displaystyle \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right]$ is true.
    Then for some s we have P(s) or for some u we have Q(u).
    In the first instance: $\displaystyle 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, $\displaystyle 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 $\displaystyle \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]$

    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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