To complete this truth table, let’s analyze each expression step-by-step. Here’s a breakdown of the symbols and operations involved:

1. p XOR q ( $p \oplus q$ ): This is true if and only if $p$ and $q$ have different truth values.
2. p XOR NOT q ( $p \oplus \sim q$ ): This is true if and only if $p$ and $\sim q$ have different truth values.
3. ( $p \oplus q$ ) OR ( $p \oplus \sim q$ ): This expression is true if either $p \oplus q$ or $p \oplus \sim q$ is true.
4. ( $p \oplus q$ ) AND ( $p \oplus \sim q$ ): This expression is true only if both $p \oplus q$ and $p \oplus \sim q$ are true.

Let’s evaluate these for each combination of $p$ and $q$ values (T = True, F = False):

Step-by-Step Evaluation:

Solution for Each Row:

1. Row 1: $p = T$, $q = T$

   • $p \oplus q = F$
   • $p \oplus \sim q = T$
   • $(p \oplus q) \lor (p \oplus \sim q) = T$
   • $(p \oplus q) \land (p \oplus \sim q) = F$

2. Row 2: $p = T$, $q = F$

   • $p \oplus q = T$
   • $p \oplus \sim q = T$
   • $(p \oplus q) \lor (p \oplus \sim q) = T$
   • $(p \oplus q) \land (p \oplus \sim q) = T$

3. Row 3: $p = F$, $q = T$

   • $p \oplus q = T$
   • $p \oplus \sim q = F$
   • $(p \oplus q) \lor (p \oplus \sim q) = T$
   • $(p \oplus q) \land (p \oplus \sim q) = F$

4. Row 4: $p = F$, $q = F$

   • $p \oplus q = F$
   • $p \oplus \sim q = F$
   • $(p \oplus q) \lor (p \oplus \sim q) = F$
   • $(p \oplus q) \land (p \oplus \sim q) = F$

Would you like further details on any step or concept?

Related Questions:

1. What is the difference between XOR and AND operators in Boolean logic?
2. How do truth tables help in understanding logical expressions?
3. Why is $p \oplus q$ true only when $p$ and $q$ are different?
4. How would this table look if we used NOR and NAND instead of OR and AND?
5. What practical applications use XOR operations?

Tip:

When analyzing truth tables, break down complex expressions step-by-step to understand each intermediate result before combining them.