To determine if the logical expression \( \sim a \vee (a \vee \sim b) \) is a **tautology**, we can evaluate its truth table. A **tautology** is a statement that is always true, regardless of the truth values of the individual variables.

The given expression is:
\[
\sim a \vee (a \vee \sim b)
\]
Let's break this down:

1. \( \sim a \) is the negation of \( a \).
2. \( a \vee \sim b \) is the disjunction (OR) of \( a \) and the negation of \( b \).
3. The whole expression is \( \sim a \vee (a \vee \sim b) \), which means we take the OR of \( \sim a \) and \( (a \vee \sim b) \).

Now, let's build the truth table for all possible values of \( a \) and \( b \).

| \( a \) | \( b \) | \( \sim a \) | \( \sim b \) | \( a \vee \sim b \) | \( \sim a \vee (a \vee \sim b) \) |
|--------|--------|---------------|---------------|----------------------|----------------------------------|
| T      | T      | F             | F             | T                    | T                                |
| T      | F      | F             | T             | T                    | T                                |
| F      | T      | T             | F             | F                    | T                                |
| F      | F      | T             | T             | T                    | T                                |

As we can see from the last column of the truth table, the expression is **true in all cases**, making it a **tautology**.

Do you want details or have any questions about this? 

Here are 5 related questions:
1. What is the difference between a tautology and a contradiction?
2. How do you find if an expression is a contradiction?
3. What role does negation play in logical expressions?
4. Can an expression be both a tautology and a contradiction?
5. How does a truth table help determine the validity of a logical expression?

**Tip:** When analyzing logical expressions, breaking them down into smaller parts makes it easier to understand their overall truth values.

Evaluate the logical expression ~a v (a v~b) to determine if it is a tautology. This problem introduces key concepts of logic, including truth tables, negation, and disjunction. Step-by-step truth table analysis provided. Suitable for students in Grades 9-12.

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