Stupid question related to proof writing.

  1. I have no idea how to type math symbols into here so it's all in the PNG attached.

    I'm probably kind of dumb for not getting this but...

    I understand that 1) & 3) are true. And the 2) is not right, as it means all x are members of F and true for P(x) when we mean all x that are members of F are true for P(x).
    But why do we use 3) instead of 4)?
     

    Attached Files:

  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,932
    Staff Emeritus
    Science Advisor

    (4) is not always a true statement. The right hand side of (4) would be true even if F were empty whereas the left hand side would not be. Notice that if x is NOT in F then "x contained in F implies P(x)" is a TRUE statement because the hypothesis is FALSE.

    matt, that was pretty much what you said. Why did you delete it?
     
    Last edited: Oct 22, 2006
  4. matt grime

    matt grime 9,395
    Science Advisor
    Homework Helper

    Cos when I looked more closely I decided that I couldn't decipher the small subscript on the LHS with any certainity.
     
  5. Thanks everyone. Sorry about the size, I attached a bigger one in this post.

    So from what I understand from reading the replies and scratching my head over the AND and IMPLIE truth tables.

    right side of 3) asserts :
    • there exist a x such that it's a member of F and true for P(x)

    right side of 4) asserts :
    1. there exist a x such that it's a member of F and true for P(x) , or
    2. there exist a x such that it's NOT a member of F and true for P(x) , or
    3. there exist a x such that it's NOT a member of F and NOT true for P(x)

    However we do not wish to state as true 2. and 3. , for it would implie that there exist a x that is NOT a member of F. As the set representing "not F" may or may not be empty.

    Anyway that's the reasoning I manage to arrive at.
     

    Attached Files:

  6. matt grime

    matt grime 9,395
    Science Advisor
    Homework Helper

    A=>B is precisely "B or not(A)".
     
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