- Evaluate each biconditional statement for rows A (\(p=T, q=T, r=T\)) and E (\(p=F, q=T, r=T\)).
- A biconditional \(x \leftrightarrow y\) is true when \(x\) and \(y\) share the same truth value.
- Statements \(q \leftrightarrow r\) and \(r \leftrightarrow q\) evaluate to True in both rows A and E.
- Final Answer: \(\boxed{3, 4}\)
Explanation
- Analysis of the Data and Problem
We analyze the truth values for the statements \(p, q, r\) in rows A and E:
- In row A, the values are \(p=T\), \(q=T\), and \(r=T\).
- In row E, the values are \(p=F\), \(q=T\), and \(r=T\).
A biconditional statement \(x \leftrightarrow y\) is true if and only if both \(x\) and \(y\) have the same truth value. We need to evaluate each of the five given options for both rows and identify which ones are true in both.
- Evaluating Statements for Row A
We evaluate each of the five proposed biconditional statements for Row A (\(p=T, q=T, r=T\)):
- \(r \leftrightarrow p \equiv T \leftrightarrow T = T\)
- \(p \leftrightarrow q \equiv T \leftrightarrow T = T\)
- \(q \leftrightarrow r \equiv T \leftrightarrow T = T\)
- \(r \leftrightarrow q \equiv T \leftrightarrow T = T\)
- \(q \leftrightarrow p \equiv T \leftrightarrow T = T\)
All five statements are true for row A.
- Evaluating Statements for Row E
We evaluate each of the five proposed biconditional statements for Row E (\(p=F, q=T, r=T\)):
- \(r \leftrightarrow p \equiv T \leftrightarrow F = F\)
- \(p \leftrightarrow q \equiv F \leftrightarrow T = F\)
- \(q \leftrightarrow r \equiv T \leftrightarrow T = T\)
- \(r \leftrightarrow q \equiv T \leftrightarrow T = T\)
- \(q \leftrightarrow p \equiv T \leftrightarrow F = F\)
From this, we see that only statements 3 (\(q \leftrightarrow r\)) and 4 (\(r \leftrightarrow q\)) are true for row E.
- Conclusion
Comparing the results from step 2 and step 3, we see that only statements 3 and 4 are true in both row A and row E. Therefore, the two correct statements are \(q \leftrightarrow r\) and \(r \leftrightarrow q\).
Examples
Understanding truth tables and biconditional statements is foundational in computer science and digital circuit design. For example, when designing an "Equality Checker" circuit, you need to determine when two signals are identical. Just as we identified that \(q\) and \(r\) are both high (True) in our specific rows, an engineer uses these logical rules to build systems that confirm when two sensors or inputs match perfectly, ensuring logical consistency in automated systems.