Please prove a statement without using the truth table that it i equivalent to another .Please clearly show the steps

To prove that two logical statements are equivalent without using a truth table, we rely on **logical identities** (rules of logic) and **transformations**. I'll walk you through an example where we prove that two logical expressions are equivalent.

### Example: 
Prove that the statement \(\neg(p \land q)\) is equivalent to \(\neg p \lor \neg q\).

This is an application of **De Morgan's Law** in logic, which states:
\[
\neg(p \land q) \equiv \neg p \lor \neg q
\]
We'll prove this step by step using logical transformations.

### Step-by-Step Proof:

#### Statement 1:
\[
\neg(p \land q)
\]
This is a negation of a conjunction. We aim to simplify and transform this into an expression that involves disjunctions (i.e., "or" statements).

#### Apply De Morgan's Law:
De Morgan's Law tells us how to distribute a negation across a conjunction or a disjunction. In this case:
\[
\neg(p \land q) \equiv \neg p \lor \neg q
\]

- **Explanation**: De Morgan's Law allows us to break apart the negation of a conjunction into a disjunction of the negations of the individual components. 

Thus, by the rule of De Morgan’s law, we have shown that:
\[
\neg(p \land q) \equiv \neg p \lor \neg q
\]

### Conclusion:
We have proven that \(\neg(p \land q)\) is equivalent to \(\neg p \lor \neg q\) by applying **De Morgan's Law**.

### Questions for further exploration:
1. Can you use De Morgan's Law to prove \(\neg(p \lor q) \equiv \neg p \land \neg q\)?
2. How would you prove \(p \lor \neg p\) is a tautology?
3. Can you show how to distribute a disjunction over a conjunction?
4. What logical identity allows us to simplify \(p \lor (p \land q)\)?
5. How can we prove that \((p \lor q) \land (p \lor \neg q)\) is a contradiction?

### Tip: 
When working with logical statements, practice identifying opportunities to apply rules like **De Morgan’s Laws**, **Distributive Laws**, and **Absorption Laws** to simplify expressions and prove equivalences.

Please prove a statement without using the truth table that it is equivalent to another. Please clearly show the steps.

Learn how to prove the equivalence of logical statements using De Morgan's Laws without relying on a truth table. This example walks through a step-by-step proof showing that ¬(p ∧ q) is equivalent to ¬p ∨ ¬q, applying key concepts in propositional logic. Ideal for students in Grades 10-12.

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