Question

1)  Recall, a truth table for a proposition involving propositional symbols p and q uses four rows for the cases p true, q true, p true, q false, p false, q true and p false, q false (in that order). For example  the outcome for p v ¬q  is  T, T, 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 ¬(¬p ∧ q), (¬p ∧ q) → false  and q → p. Of these, q → 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) FFFF

b) FFFT

c) FFTF

d) FFTT

e) FTFF

f) FTFT

g) FTTF

h) FTTT

i) TFFF

j) TFFT

k) TFTF

l) TFTT

m) TTFF

n) TTTF

o) TTTT

You may only use symbols from the set p, q, → , ∧ , ∨, (,  ),  ↔ , false,  true, ¬ .  Each of those count 1 toward the length of the expression. (Note:  falseand 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 p or q in them!

Answer 1

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