Truth-functional completeness

Question

Let the statement $?PQR$ be determined by the following truth-table.

P   Q   R   ?PQR
T   T   T   T
T   T   F   F
T   F   T   F
T   F   F   T
F   T   T   T
F   T   F   T
F   F   T   F
F   F   F   T

1. After ‘Answer:’ below, give a logically equivalent sentence of ?PQR in FOL. But here’s the catch: you may only use the Boolean connectives (i.e. ¬, Ʌ, and V) in the sentence you give. I am trying to figure this assignment out; but cannot figure out the formula for determining the sentence. Any ideas?

Answer 1

HINT:

For every line whose value is $\tt T$, write a sentence describe the assignment to the variables, e.g. $P\land Q\land R$ describes the first line of the table.

Then take the disjunction of these sentences, and show that the result has exactly this truth table.

You might want to simplify that result.

Answer 2

@AsafKaragila, being a nice guy, has given you a hint, and patiently explained more in his comments. Being not such a nice guy, can I point out that your question rather suggests that you haven't been to the library and done your basic groundwork? More or less any elementary logic text will have a section, under the title "expressive completeness" or some such, which will explain this idea. Faced with such a very basic question like this, your first reaction should be to check out two or three texts about the relevant principles at stake, till you find one whose explanations give you that "Aha! Got it!!" feeling.