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

Concept: The implication formula \[ A \Rightarrow B \] is logically equivalent to \[ \neg A \lor B. \] Hence, the given proposition can be simplified and then analyzed using the truth table principle.

Step 1: Simplify the given proposition.
Given, \[ (\neg p \lor q)\Rightarrow r. \] Using \[ A\Rightarrow B \equiv \neg A \lor B, \] we obtain \[ (\neg p\lor q)\Rightarrow r = \neg(\neg p\lor q)\lor r. \] Applying De Morgan's Law, \[ = (p\land \neg q)\lor r. \]

Step 2: Determine when the statement is false.
The expression \[ (p\land \neg q)\lor r \] is false only when both parts are false. Thus, \[ r=F \] and \[ p\land \neg q=F. \]

Step 3: Count the false cases.
When \(r=F\), the possible values of \((p,q)\) are: \[ (T,T),\quad(F,T),\quad(F,F) \] for which \[ p\land\neg q=F. \] Hence the number of false cases is \[ 3. \]

Step 4: Calculate the number of true cases.
Since there are three variables, \[ \text{Total assignments}=2^3=8. \] Therefore, \[ \text{True assignments} = 8-3 = 5. \] \[ \boxed{5} \]