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Diccrete Math problem: finding a proposition given a specific truth table

  1. Sep 30, 2012 #1
    I've been working at this problem for a while and it seems that there should be an easier more systematic way of solving it. Here it is:

    Find a proposition using only p, q, ¬ and the connective ∧ with the given truth table.

    p q ?
    T T F
    T F F
    F T T
    F F T

    I know of a systematic approach to creating propositions but includes the use of OR. Is there a systematic way of solving this only using ANDs?
  2. jcsd
  3. Sep 30, 2012 #2
    Just read off the ones where the truth table evaluates to "T", namely:
    F T T
    F F T
    This results in:
    (not p and q) (from the F T T line)
    (not p and not q) (from the F F T line)

    join them with "or":
    (not p and q) or (not p and not q)

    simplify if possible, using the distributive rule in this case:
    (not p) and (q or not q)
    q or not q = T
    (not p) and T = not p

    final result:
    not p
  4. Sep 30, 2012 #3
    Thanks this is very helpful. It's actually the way I originally went about solving the problem but the book gives another answer:

    ¬(p ∧ ¬q) ∧ ¬(p ∧ q)

    Probably both answers are acceptable, but I wonder how I could get the book's answer from your original answer, (¬p ∧ q) ∨ (¬p ∧ ¬q). There must be a way to convert it since they are logically equivalent.
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