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

## Main Question or Discussion Point

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?

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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

• InfZero
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.