Question-15

Let \(p\) and \(q\) be any two propositions. Consider the following propositional statements:

• \(S_{1} :p\rightarrow q\)
• \(S_{2} :\neg p\land q\)
• \(S_{3} :\neg p\lor q\)
• \(S_{4} :\neg p\lor \neg q\)

where \(\land\) denotes conjunction (AND operation), \(\lor\) denotes disjunction (OR operation), and \(\neg\) denotes negation (NOT operation). Which one of the following operations is correct. (Note: \(\equiv\) denotes logical equivalence)

• \(S_{1} \equiv S_{3}\)

• \(S_{2} \equiv S_{3}\)

• \(S_{2} \equiv S_{4}\)

• \(S_{1} \equiv S_{4}\)

NoteAnswer

• \(S_{1} \equiv S_{3}\)

• \(S_{2} \equiv S_{3}\)

• \(S_{2} \equiv S_{4}\)

• \(S_{1} \equiv S_{4}\)

NoteSolution

Let us draw the truth table:

\[ \begin{array}{|c|c|c|c|c|c|} \hline p & q & S_{1} & S_{2} & S_{3} & S_{4}\\ \hline 0 & 0 & 1 & 0 & 1 & 1\\ \hline 0 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 0 & 0 & 0 & 0 & 1\\ \hline 1 & 1 & 1 & 0 & 1 & 0\\ \hline \end{array} \]

Two statements are equivalent if their truth tables are identical. From this, we see that \(S_{1} \equiv S_{3}\).