https://r2.erweima.ai/mathbot/image/1726532887108wlwhplrl.jpg

The image contains a logical expression, which is:

1. \[(p \land q) \to r \land \neg q\]

This is a conditional statement that involves conjunctions and negations. Let's break it down and analyze it step by step:

### Step 1: Understand the symbols
- \( p \) and \( q \): Propositional variables.
- \( \land \): Logical AND (conjunction).
- \( \neg \): Logical NOT (negation).
- \( \to \): Logical implication (if-then).

### Step 2: Parenthesize the expression
To avoid ambiguity, let's clearly separate different parts of the expression:
\[ \left( (p \land q) \to r \right) \land \neg q \]

This expression means:
- The first part, \( (p \land q) \to r \), is an implication: "If both \( p \) and \( q \) are true, then \( r \) is true."
- The second part, \( \neg q \), negates \( q \), meaning "not \( q \)."
- The logical AND (\( \land \)) combines the two statements.

### Step 3: Truth Table Construction
We can evaluate the truth table for this expression by listing all possible truth values of \( p \), \( q \), and \( r \), and then evaluating the expression:

| \( p \) | \( q \) | \( r \) | \( p \land q \) | \( (p \land q) \to r \) | \( \neg q \) | \( \left( (p \land q) \to r \right) \land \neg q \) |
|--------|--------|--------|--------------|-------------------|------------|-------------------------------------------|
| T      | T      | T      | T            | T                 | F          | F                                         |
| T      | T      | F      | T            | F                 | F          | F                                         |
| T      | F      | T      | F            | T                 | T          | T                                         |
| T      | F      | F      | F            | T                 | T          | T                                         |
| F      | T      | T      | F            | T                 | F          | F                                         |
| F      | T      | F      | F            | T                 | F          | F                                         |
| F      | F      | T      | F            | T                 | T          | T                                         |
| F      | F      | F      | F            | T                 | T          | T                                         |

### Final Answer:
The truth table helps us evaluate when the expression is true. The expression \( \left( (p \land q) \to r \right) \land \neg q \) is **true** in the following cases:
1. \( p = T \), \( q = F \), \( r = T \)
2. \( p = T \), \( q = F \), \( r = F \)
3. \( p = F \), \( q = F \), \( r = T \)
4. \( p = F \), \( q = F \), \( r = F \)

Do you want more details or have any other questions?

### Related Questions:
1. How do conjunction and implication differ in logic?
2. How is a truth table for negation built?
3. Can you create a similar problem with more propositional variables?
4. What is the difference between a tautology and a contradiction in propositional logic?
5. How does De Morgan's Law apply to this problem?

### Tip:
For complex logical expressions, always parenthesize and simplify terms step by step before building the truth table!

This problem involves analyzing the logical expression (p ∧ q) → r ∧ ¬q using truth tables and key concepts in propositional logic. Suitable for students in Grades 10-12, it covers conjunction, implication, and negation.

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