Let's go through the truth table in your image step by step to complete it.

The given expression is $p \oplus (p \lor q)$, where:

• $p \oplus q$ represents the exclusive OR (XOR), which is true if either $p$ or $q$ is true, but not both.
• $p \lor q$ is the logical OR, which is true if at least one of $p$ or $q$ is true.

You have the following partial truth table:

Step 1: Complete $p \lor q$ (already done in the table)

• When $p = T$ and $q = T$, $p \lor q = T$.
• When $p = T$ and $q = F$, $p \lor q = T$.
• When $p = F$ and $q = T$, $p \lor q = T$.
• When $p = F$ and $q = F$, $p \lor q = F$.

This part is already filled in the table.

Step 2: Calculate $p \oplus (p \lor q)$

This is the XOR operation between $p$ and $p \lor q$.

• $p \oplus (p \lor q)$ is true if $p$ and $p \lor q$ have different truth values.

Final Answer:

• For $p = T$ and $q = T$, $p \oplus (p \lor q) = F$.
• For $p = T$ and $q = F$, $p \oplus (p \lor q) = F$.
• For $p = F$ and $q = T$, $p \oplus (p \lor q) = T$.
• For $p = F$ and $q = F$, $p \oplus (p \lor q) = F$.

Would you like more details on XOR operations or any other parts of this problem?

Here are five related questions to deepen your understanding:

1. What is the difference between XOR and OR in truth tables?
2. How would you build a truth table for $p \land q$ (AND operation)?
3. What does the logical equivalence $p \oplus q \equiv \neg(p \iff q)$ mean?
4. How can you express $p \oplus q$ using only AND, OR, and NOT operations?
5. What are the common uses of XOR in digital logic design?

Tip: XOR is useful for detecting differences between two inputs, often used in error detection and correction algorithms!