Use a truth table to determine whether the statement is a​ tautology, a​ self-contradiction, or neither.
left parenthesis p left right arrow q right parenthesis left right arrow left bracket left parenthesis q right arrow p right parenthesis logical and left parenthesis p right arrow q right parenthesis right bracket(p↔q)↔[(q→p)∧(p→q)]
Question content area bottom
Part 1
Complete the truth table.
Part 2Part 3Part 4
p
q
p left right arrow qp↔q
q right arrow pq→p
p right arrow qp→q
left parenthesis q right arrow p right parenthesis logical and left parenthesis p right arrow q right parenthesis(q→p)∧(p→q)
left parenthesis p left right arrow q right parenthesis left right arrow left bracket left parenthesis q right arrow p right parenthesis logical and left parenthesis p right arrow q right parenthesis right bracket(p↔q)↔[(q→p)∧(p→q)]
T
T
enter your response here
enter your response here
enter your response here
enter your response here
enter your response here
T
F
enter your response here
enter your response here
enter your response here
enter your response here
enter your response here
F
T
enter your response here
enter your response here
enter your response here
enter your response here
enter your response here
F
F
enter your response here
enter your response here
enter your response here
enter your response here
enter your response here
Part 5
Is the statement left parenthesis p left right arrow q right parenthesis left right arrow left bracket left parenthesis q right arrow p right parenthesis logical and left parenthesis p right arrow q right parenthesis right bracket(p↔q)↔[(q→p)∧(p→q)] a​ tautology, a​ self-contradiction, or​ neither?

Tautology

Neither

​Self-contradiction

We can use a truth table to analyze the statement:

\[
(p \leftrightarrow q) \leftrightarrow [(q \rightarrow p) \land (p \rightarrow q)]
\]

Here's how we can approach it step by step.

### Step 1: Truth table structure

We'll need to evaluate each component:
1. \( p \)
2. \( q \)
3. \( p \leftrightarrow q \) (biconditional)
4. \( q \rightarrow p \) (implication)
5. \( p \rightarrow q \) (implication)
6. \( (q \rightarrow p) \land (p \rightarrow q) \) (conjunction of implications)
7. \( (p \leftrightarrow q) \leftrightarrow [(q \rightarrow p) \land (p \rightarrow q)] \)

We'll evaluate the main logical expression in each case.

### Truth Table

| \( p \) | \( q \) | \( p \leftrightarrow q \) | \( q \rightarrow p \) | \( p \rightarrow q \) | \( (q \rightarrow p) \land (p \rightarrow q) \) | \( (p \leftrightarrow q) \leftrightarrow [(q \rightarrow p) \land (p \rightarrow q)] \) |
|--------|--------|--------------------------|-----------------------|-----------------------|--------------------------------------------------|---------------------------------------------------------------------------------------|
| T      | T      | T                        | T                     | T                     | T                                                | T                                                                                     |
| T      | F      | F                        | T                     | F                     | F                                                | T                                                                                     |
| F      | T      | F                        | F                     | T                     | F                                                | T                                                                                     |
| F      | F      | T                        | T                     | T                     | T                                                | T                                                                                     |

### Step 2: Analyze the final column

The final column represents the overall truth value of the statement for each combination of truth values of \( p \) and \( q \).

### Step 3: Conclusion

Since the last column has all "T" (True) values, this statement is **always true**, which means the given statement is a **tautology**.

---

**Answer: Tautology**

Do you want further details on any specific step? Here are some related questions for practice:

1. What is the definition of a biconditional statement \( p \leftrightarrow q \)?
2. How does an implication \( p \rightarrow q \) work in logic?
3. Can you explain why \( (q \rightarrow p) \land (p \rightarrow q) \) is equivalent to \( p \leftrightarrow q \)?
4. How do truth tables help in determining if a statement is a tautology or a contradiction?
5. Could a compound statement involving biconditionals and implications ever be a contradiction?

**Tip**: In a biconditional \( p \leftrightarrow q \), both \( p \) and \( q \) must have the same truth value for the statement to be true.

Use a truth table to determine whether the statement is a tautology, a self-contradiction, or neither.
(p↔q)↔[(q→p)∧(p→q)]

Learn how to use a truth table to determine whether the logical statement (p↔q)↔[(q→p)∧(p→q)] is a tautology, self-contradiction, or neither. This step-by-step guide covers key concepts in logic such as biconditional, implication, and conjunction. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation