Question

The binary operation $\square$ is defined as follows:

P	Q	P$\square$Q
T	T	T
T	F	T
F	T	F
F	F	T

Which one of the following is equivalent to P $\vee$ Q?

• A. $\neg Q\square \neg P$
• B. $P\square \neg Q$
• C. $\neg P\square Q$
• D. $\neg P\square \neg Q$

Answer 1

The correct option is B $P\square \neg Q$

The given table can be converted into boolean function by adding minterms corresponding to true rows.
$\therefore$

P	Q	P$\square$Q
T	T	T
T	F	T
F	T	F
F	F	T

Since there is only one false in the above truth table. we can represent the function P $\square$ Q more efficientely. in conjunctive normal form. Translates P$\square$Q=P+ Q' (the max-term corresponding to the third row, where the function is false).
Now, we can easily translate the choices into boolean algebra as follows:
Choice (a) $\neg Q\square \neg P\equiv Q^{\prime }\square P^{\prime }\equiv Q^{\prime }+P$
Choice (b) $P\square \neg Q\equiv P\square Q^{\prime }\equiv P+Q$
Choice (c) $\neg P\square Q\equiv P^{\prime }\square Q\equiv P^{\prime }+Q^{\prime }$
Choice (d) $\neg P\square \neg Q\equiv P^{\prime }\square Q^{\prime }\equiv P^{\prime }+Q$
As we can clearly see only choice (b) $P\square \neg Q$ is equivalent to P + Q.