Question

The number of ordered tuples \((p,q,r)\) in truth table for which the statement \((\neg p \lor q)\Rightarrow r\) is true is:

• A. \(3\)
• B. \(4\)
• C. \(2\)
• D. \(5\)

Hint:

For propositions involving implication, first convert \(A\Rightarrow B\) into \(\neg A\lor B\). Then count the false cases, since implication is false only when the antecedent is true and the consequent is false.

Answer 1

The Correct Option is D

Step 1: Rewrite the implication.
Use $A\Rightarrow r\equiv \neg A\lor r$ with $A=\neg p\lor q$. So the statement is $\neg(\neg p\lor q)\lor r$.
Step 2: Apply De Morgan.
$\neg(\neg p\lor q)=p\land\neg q$, so the statement simplifies to $(p\land\neg q)\lor r$.
Step 3: Find when it is FALSE.
An OR is false only when both sides are false: we need $r=F$ and $p\land\neg q=F$.
Step 4: Count the false rows.
Fix $r=F$. Then $p\land\neg q=F$ holds for $(p,q)=(T,T),(F,T),(F,F)$, three combinations. So there are $3$ false rows total.
Step 5: Count the total rows.
With three variables $p,q,r$ there are $2^3=8$ rows in the truth table.
Step 6: Subtract to get the true rows.
True rows $=8-3=5$, option 4.
\[ \boxed{5} \]