Simplification of Logic

  1. 1. The problem statement, all variables and given/known data

    Simplify the following compound statements (give a smallest formula equivalent to each of them). State which logical identities you used at each step
    (a) (p→q)↔(q→p)
    (b) ¬(p∧q)→(q→(p∨q))


    2. Relevant equations

    n/a

    3. The attempt at a solution

    (a)
    ( p → q ) ↔ ( q → p )
    ¬( p \/ q ) ↔ ¬( q \/ p )
    ¬( p \/ q )


    (b)
    ¬(p /\ q) → (q → (p \/ q)
    (¬ p \/ ¬ q) → (q → (p \/ q)
    ¬(¬p \/ ¬q) \/ (q → (p \/ q)
    (p \/ q) \/ (¬q \/ p \/ q)
    (p \/ q) \/ (q \/ p)
    (p \/ q)

    So these are my attempts, My question stems from part B, in these two lines

    ¬(¬p \/ ¬q) \/ (q → (p \/ q)
    (p \/ q) \/ (¬q \/ p \/ q)

    With the second right ward facing arrow. You have to apply De Morgan's law in order to get rid of it, but when applying the not to equation does it go to the right of the bracket like so

    (p \/ q) \/ ¬(q \/ p \/ q)

    Or the way i have it

    (p \/ q) \/ (¬q \/ p \/ q)

    I like it the way i have it, because it works so much nicer, but i'd like to be certain. Help
     
  2. jcsd
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