Suppose p and q are true and r is false. What is the truth value of p ^ (~pVq V ~ r)

We are asked to find the truth value of the expression:

\[
p \land (\neg p \lor q \lor \neg r)
\]

Given the truth values:

- \( p \) is **true**,
- \( q \) is **true**,
- \( r \) is **false**.

### Step 1: Simplify the inner expression \( \neg p \lor q \lor \neg r \)

- \( \neg p \): Since \( p \) is true, \( \neg p \) (not \( p \)) is **false**.
- \( q \): Given that \( q \) is true, it remains **true**.
- \( \neg r \): Since \( r \) is false, \( \neg r \) (not \( r \)) is **true**.

So, the expression \( \neg p \lor q \lor \neg r \) becomes:

\[
\text{false} \lor \text{true} \lor \text{true} = \text{true}
\]

### Step 2: Evaluate the whole expression \( p \land (\neg p \lor q \lor \neg r) \)

Now we have:

\[
p \land \text{true}
\]

Since \( p \) is **true**, the entire expression becomes:

\[
\text{true} \land \text{true} = \text{true}
\]

Thus, the truth value of \( p \land (\neg p \lor q \lor \neg r) \) is **true**.

Would you like more details on this? Here are some related questions for practice:

1. What happens if \( r \) is true instead of false?
2. How do truth tables help in evaluating such logical expressions?
3. What is the truth value of \( \neg p \lor q \) if both \( p \) and \( q \) are false?
4. What is the general method to evaluate expressions involving \( \land \) and \( \lor \)?
5. Can the expression \( \neg p \lor q \lor \neg r \) be simplified further in some cases?

**Tip:** Always simplify the innermost expressions first when evaluating complex logical statements!

Explore how to find the truth value of the logical expression p AND (NOT p OR q OR NOT r) given that p and q are true, and r is false. This problem involves understanding Boolean algebra, logical connectives, and simplification techniques, making it suitable for high school students in Grades 10-12.

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