- Evaluate the expression \((p \land q) \lor (p \land r)\) for each row using the logical OR property, which is true if either component is true.
- Row A results in \((T \lor T) = T\).
- Row B results in \((T \lor F) = T\).
- Row C results in \((F \lor T) = T\).
- Rows D through H all result in \((F \lor F) = F\).
- The expression is true for rows A, B, and C, which corresponds to option \(\boxed{c}\).
Explanation
- Analyzing the Logical Expression
To determine which rows result in the logical expression \((p \land q) \lor (p \land r)\) being true, we analyze the provided truth table for each row from A to H. The expression \((p \land q) \lor (p \land r)\) uses the logical disjunction operator (OR, denoted by \(\lor\)), which evaluates to true if at least one of its components, either \((p \land q)\) or \((p \land r)\), is true.
- Evaluating the Truth Table Rows
We evaluate the expression \((p \land q) \lor (p \land r)\) for each row of the provided truth table:
- Row A: \(p \land q = T\) and \(p \land r = T\). Since at least one is true, \((T \lor T) = T\).
- Row B: \(p \land q = T\) and \(p \land r = F\). Since \(p \land q\) is true, \((T \lor F) = T\).
- Row C: \(p \land q = F\) and \(p \land r = T\). Since \(p \land r\) is true, \((F \lor T) = T\).
- Row D: \(p \land q = F\) and \(p \land r = F\). Since both are false, \((F \lor F) = F\).
- Row E: \(p \land q = F\) and \(p \land r = F\). Since both are false, \((F \lor F) = F\).
- Row F: \(p \land q = F\) and \(p \land r = F\). Since both are false, \((F \lor F) = F\).
- Row G: \(p \land q = F\) and \(p \land r = F\). Since both are false, \((F \lor F) = F\).
- Row H: \(p \land q = F\) and \(p \land r = F\). Since both are false, \((F \lor F) = F\).
- Conclusion
From our evaluation, the logical expression \((p \land q) \lor (p \land r)\) is true for rows A, B, and C. Comparing this to the given choices:
AA and B
BB, C, and E
CA, B, and C
DB and E
The correct choice is c. Thus, the rows representing when the expression is true are A, B, and C. \(\boxed{c}\)
Examples
Logical expressions are the backbone of decision-making systems in computer science and engineering. For example, in a database query, you might want to retrieve records of customers who are "Active" (p) AND have an "Account Balance > \$0" (q), OR retrieve records of customers who are "Active" (p) AND "Have an outstanding loan" (r). The expression \((p \land q) \lor (p \land r)\) ensures that your search covers both conditions effectively, helping you filter data efficiently for targeted marketing or account management.