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The question asks to:

1. Define **tautology** and **contradiction**.
2. Prove that:
   - \( a) \ \neg p \lor p \) is a contradiction.
   - \( b) \ \neg p \land (\neg p) \) is a tautology.

### Definitions:
- **Tautology**: A logical formula that is always **true** regardless of the truth values of its components.
- **Contradiction**: A logical formula that is always **false** regardless of the truth values of its components.

### Now, let's analyze the statements:

#### a) \(\neg p \lor p\)
This is a well-known logical expression, also called the **law of excluded middle**. In logical terms:
- When \( p \) is **true**, \(\neg p\) is **false**, and so \( \neg p \lor p = false \lor true = true \).
- When \( p \) is **false**, \(\neg p\) is **true**, and so \( \neg p \lor p = true \lor false = true \).

Thus, \( \neg p \lor p \) is **always true**, meaning it is a **tautology**, not a contradiction as labeled in part a.

#### b) \(\neg p \land \neg p\)
This simplifies to just \( \neg p \), because \( \neg p \land \neg p \) is the same as saying \( \neg p \), and it holds depending on the truth value of \( p \).
- When \( p \) is **true**, \(\neg p\) is **false**.
- When \( p \) is **false**, \(\neg p\) is **true**.

Therefore, \( \neg p \land \neg p \) is not always true or always false; it depends on \( p \)'s value, so it is **not** a tautology, but rather a **contingency**.

It seems like there may be an error in the labeling or intention of the question. Let me know if you'd like further clarification or more details.

Would you like me to provide more explanations? Here are five follow-up questions that might help:

1. What are examples of tautologies in logic?
2. How does the law of excluded middle work in propositional logic?
3. Can contradictions be derived from any tautology?
4. How would you construct a truth table for \( \neg p \lor p \)?
5. What is the difference between a tautology and a contingency?

**Tip**: Whenever confused about logical formulas, constructing a truth table is an effective method to validate whether the statement is a tautology, contradiction, or contingency.

Define tautology and contradiction. Show that a) ~(p ∨ ~p) is a contradiction. b) ~(p ∧ ~p) is a tautology.

This question explores logical statements in propositional logic, asking to prove that ~(p ∨ ~p) is a contradiction and ~(p ∧ ~p) is a tautology. Concepts such as the law of excluded middle and law of non-contradiction are covered. Suitable for Grades 11-12.

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