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Math Help - logically equivalent

  1. #1
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    logically equivalent

    Determine whether All X ( P(X) If only if Q(X)) and All X P(X) If only if All X Q(X) are logically equivalent.


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  2. #2
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    Quote Originally Posted by questionboy View Post
    Determine whether All X ( P(X) If only if Q(X)) and All X P(X) If only if All X Q(X) are logically equivalent.


    Thank you very much

    Seems to be you want to check whether \forall x\left(P(x)\Longleftrightarrow Q(x)\right)\equiv \left(\forall x\,P(x)\Longleftrightarrow \forall x\,Q(x)\right) .

    Well, let \mathbb{Z} be the universe from where we get our objects ,and let P(x) = the absolute value of x is x itself (i.e., P(x) iff |x|=x ,and let Q(x)= x is a non-negative integer (i.e., Q(x) iff x\geq 0) , then:

    \forall x\left(P(x)\Longleftrightarrow Q(x)\right) means: for any integer x, |x|=x\,\,\,iff\,\,\,x\leq 0 , which is true, whereas

    \left(\forall x\,P(x)\Longleftrightarrow \forall x\,Q(x)\right) means: for any integer x we have |x|=x\,\,\,iff for any integer x, x\geq 0 , which is false.

    If I didn't commit some logical mistake above () then the answer is: no, they aren't logical equivalent.

    Tonio
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  3. #3
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    No. Consider P(x) = "x=0", and Q(x)="x>0"
    Clearly For all x, P(x) iff Q(x) is false. But, "for all x, P(x)" is false, and "for all x, Q(x)" is false. Then "for all x, P(x) iff for all x, Q(x)" is true.

    The converse is true though, because assume "for all x, P(x) iff Q(x)". Then if there there is a x where P(x) is false then Q(x) is also false, so "for all x, P(x) iff for all x, Q(x)" is true as each universal is false. Otherwise, P(x) is always true, as is Q(x); result follows.
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  4. #4
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    Thank you very much!

    I am wondering why this is false

    means: for any integer x we have for any integer x, , which is false
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  5. #5
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    Quote Originally Posted by questionboy View Post
    Thank you very much!

    I am wondering why this is false

    means: for any integer x we have for any integer x, , which is false

    Because it is not true that for any integer |x|=x iff for every integer x, x>=0...for example , |7|=7 but it is NOT true that x >=0 for every integer...say, -1 < 0...

    Tonio
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