Write 'T' for True and 'F' for False.
If `(p ^^ q) implies r` in this statement '`implies`' is a principal connective.

Answer

To determine whether the statement "if (p ∧ q) implies r, in this statement 'implies' is a principal connective" is true or false, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the definition of a principal connective**: A principal connective is defined as a connective that is not within the scope of any other connective. This means it is the main connective that determines the overall truth value of the statement.

**Hint**: Remember that a principal connective is the one that has the highest precedence in a logical expression.

2. **Identify the components of the statement**: The statement given is "(p ∧ q) implies r". Here, we have two logical connectives: conjunction (∧) and implication (→).

**Hint**: Break down the statement into its components to analyze the connectives involved.

3. **Determine the precedence of the connectives**: The precedence order of logical connectives is as follows:
- Negation (¬) - highest precedence
- Conjunction (∧) - second highest
- Disjunction (∨) - third
- Implication (→) - fourth
- Biconditional (↔) - lowest

**Hint**: Knowing the precedence helps in understanding which connective is evaluated first in a logical expression.

4. **Analyze the expression (p ∧ q) implies r**: In this expression, the conjunction (p ∧ q) is evaluated first due to its higher precedence over the implication (→). Therefore, the implication connects the result of (p ∧ q) to r.

**Hint**: Look at how the connectives interact with each other based on their precedence.

5. **Identify the principal connective**: Since the implication (→) is the outermost connective that connects the entire expression, it is considered the principal connective.

**Hint**: The principal connective is the one that determines the overall truth value of the entire statement.

6. **Conclusion**: Since the statement claims that "implies" is a principal connective and we have established that it is indeed the principal connective in the expression (p ∧ q) implies r, the statement is true.

**Hint**: Verify your conclusion by checking if the connective identified meets the criteria of being a principal connective.

### Final Answer:
The truth value of the statement is **T** (True).