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Show if this is a Tautology

  1. Apr 25, 2009 #1
    1. The problem statement, all variables and given/known data

    Determine if the following is a tautology:

    ((p → q) Ʌ (q → p) → (p Ʌ q)

    I don´t know how to show this. Can somebody pls show me all the steps
     
  2. jcsd
  3. Apr 25, 2009 #2

    Cyosis

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    Make the following table:

    p|q|p → q|q → p|(p → q) Ʌ (q → p)|p Ʌ q|((p → q) Ʌ (q → p) → (p Ʌ q)
    T|T|
    T|F|
    F|T|
    F|F|

    Now finish this table, if the last column yields true for all possible values for p and q then ((p → q) Ʌ (q → p) → (p Ʌ q) is a tautology.
     
  4. Apr 25, 2009 #3
    im sorry but im lost. this is very tricky. i dont understand this table
     
  5. Apr 25, 2009 #4

    Cyosis

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    I have basically chopped your original expression into chunks. Every column of the table has a term of the original expression in it and I have used | to separate the columns. The last column has the entire expression in it.

    Lets finish the first row:
    p and q are true so q->p is true and p->q thus (p → q) Ʌ (q → p) is true. On the other hand we have p Ʌ q which is true so now we have all components that we want. So we can conclude, since (p → q) Ʌ (q → p) is true and (p Ʌ q) is true, ((p → q) Ʌ (q → p) → (p Ʌ q) must be true.

    Now try to work your way through the other values of initial p and q. Note that I put all possible combinations in the first two columns.
     
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