To simplify the expression \((\sim p \lor q) \Rightarrow (\sim p \lor q)\), we can follow these steps:

### Step 1: Recognize the implication
The implication \(A \Rightarrow B\) is logically equivalent to \(\sim A \lor B\). Applying this to the given expression:

\[
(\sim p \lor q) \Rightarrow (\sim p \lor q) \equiv \sim(\sim p \lor q) \lor (\sim p \lor q)
\]

### Step 2: Apply De Morgan's Law
We need to negate \(\sim p \lor q\) as part of the equivalence transformation:
\[
\sim(\sim p \lor q) \equiv (\sim \sim p \land \sim q) \equiv (p \land \sim q)
\]

### Step 3: Substitute into the expression
Now substitute this back into the implication expression:
\[
(p \land \sim q) \lor (\sim p \lor q)
\]

### Step 4: Simplify using distribution
To simplify this further, we distribute the terms:
\[
(p \lor \sim p) \lor (p \lor q) \lor (\sim q \lor \sim p) \lor (\sim q \lor q)
\]

Using the tautologies \(p \lor \sim p = \text{True}\) and \(\sim q \lor q = \text{True}\), we get:

\[
\text{True} \lor (p \lor q) \lor \text{True}
\]

Since \(\text{True} \lor\) anything is always \(\text{True}\), the entire expression simplifies to:

\[
\boxed{\text{True}}
\]

Thus, the simplified form of \((\sim p \lor q) \Rightarrow (\sim p \lor q)\) is **True**.

Would you like any more details or have questions?

Here are 5 related questions to consider:
1. What is the difference between a tautology and a contradiction?
2. How does De Morgan's law work with negations?
3. What is the truth table for logical implications?
4. Can all logical implications be simplified to True or False?
5. How does logical equivalence relate to algebraic identities?

**Tip:** Logical equivalences can often be simplified by recognizing common tautologies and contradictions in their expressions.

Learn how to simplify the logical implication (∼p ∨ q) ⇒ (∼p ∨ q) using propositional logic. This detailed solution includes key concepts such as De Morgan's Laws and logical tautologies. Suitable for undergraduate students studying discrete mathematics or introductory logic.

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