Recall, a truth table for a proposition involving propositional symbols p and q uses four rows for the cases p true, a true, p true, q false, p false, q true and p false, q false (in that order). for example the outcome for pv-g is tt, f, t since the expression is only false when q is true but p is false. of course, we have the same outcome for any logically equivalent proposition including ), (-paq) → false and q- p. of these, 9-p clearly reduces the number of symbols to a bare minimum. find "minimal expressions for the other 15 possible outcomes, which are listed below: a) b) ffft c) fftf d) fftt e) ftff f) ftft g) fttf h) fttt d) tfff d) tfft k) tftf d) tett m) ttff n) ttte o) you may only use symbols from the set p, q, —,1,v, (, , , false, true, - . each of those count 1 toward the length of the expression. (note: false and true still count as single symbols, even though they have multiple letters.) in some of the answers, your expression might have just p in it and not q or vice versa. of course, there are also two answers that don't have either por q in them! type or paste your answer in the write submission box. if you are are typing you can use the symbol - in place of - and --> in place of —. also, ^ (above the '6' on your keyboard) works for and the letter v works for v.