Key Concepts and Formulas

• Truth Tables: A truth table is a table that shows all possible truth values of a logical expression based on the truth values of its variables.
• Logical Connectives:
  • $p \vee q$ (OR): True if either $p$ or $q$ or both are true. False only if both $p$ and $q$ are false.
  • $p \wedge q$ (AND): True only if both $p$ and $q$ are true. False otherwise.
  • $p \Rightarrow q$ (Implication): False only if $p$ is true and $q$ is false. True otherwise.

Step-by-Step Solution

Step 1: Construct the truth table for all possible combinations of p, q, and r.

Since each of $p, q, r$ can be either True (T) or False (F), there are $2^3 = 8$ possible combinations of their truth values. We will list all these combinations in the first three columns of the truth table.

Step 2: Evaluate $p \vee q$ for all combinations.

The truth value of $p \vee q$ is T if either $p$ or $q$ (or both) is T. It is F only when both $p$ and $q$ are F.

Step 3: Evaluate $p \vee r$ for all combinations.

The truth value of $p \vee r$ is T if either $p$ or $r$ (or both) is T. It is F only when both $p$ and $r$ are F.

Step 4: Evaluate $(p \vee q) \wedge (p \vee r)$ for all combinations.

The truth value of $(p \vee q) \wedge (p \vee r)$ is T only if both $p \vee q$ and $p \vee r$ are T. It is F otherwise.

Step 5: Evaluate $q \vee r$ for all combinations.

The truth value of $q \vee r$ is T if either $q$ or $r$ (or both) is T. It is F only when both $q$ and $r$ are F.

Step 6: Evaluate $(p \vee q) \wedge (p \vee r) \Rightarrow (q \vee r)$ for all combinations.

The truth value of the implication $(p \vee q) \wedge (p \vee r) \Rightarrow (q \vee r)$ is F only if $(p \vee q) \wedge (p \vee r)$ is T and $q \vee r$ is F. Otherwise, it is T.

The complete truth table is shown below:

$$\begin{array}{|c|c|c|c|c|c|c|c|} \hline p & q & r & p \vee q & p \vee r & (p \vee q) \wedge (p \vee r) & q \vee r & (p \vee q) \wedge (p \vee r) \Rightarrow (q \vee r) \\ \hline T & T & T & T & T & T & T & T \\ \hline T & T & F & T & T & T & T & T \\ \hline T & F & T & T & T & T & T & T \\ \hline T & F & F & T & T & T & F & F \\ \hline F & T & T & T & T & T & T & T \\ \hline F & T & F & T & F & F & T & T \\ \hline F & F & T & F & T & F & T & T \\ \hline F & F & F & F & F & F & F & T \\ \hline \end{array}$$

Step 7: Count the number of triplets for which the implication is True.

From the last column of the truth table, we can see that the implication $(p \vee q) \wedge (p \vee r) \Rightarrow (q \vee r)$ is True in 7 cases.

Common Mistakes & Tips

• Remember the truth tables of the logical connectives accurately, especially for implication ($\Rightarrow$). The implication $p \Rightarrow q$ is only false when $p$ is true and $q$ is false.
• Be systematic when constructing the truth table to avoid missing any combinations of truth values for $p, q,$ and $r$.
• The expression can be simplified before constructing the truth table, but for JEE, it's safer to use the truth table, especially if you are prone to making algebraic errors with Boolean algebra.

Summary

We constructed a truth table for the given logical expression $(p \vee q) \wedge (p \vee r) \Rightarrow (q \vee r)$ for all possible combinations of truth values of $p, q,$ and $r$. By examining the last column of the truth table, we counted the number of triplets $(p, q, r)$ for which the expression is True. We found that there are 7 such triplets.

Final Answer

The final answer is $\boxed{7}$.