Question
The conditional statement is :
Options
Solution
Key Concepts and Formulas
- Conditional Statement: The conditional statement is false only when is true and is false. Otherwise, it is true.
- Tautology: A tautology is a compound statement that is always true, regardless of the truth values of its individual components.
- Truth Table: A truth table is a table that shows all possible combinations of truth values for a compound statement.
Step-by-Step Solution
Step 1: Define the component statements
Let's define the following statements to simplify the given expression:
The given statement can then be rewritten as:
Step 2: Construct the truth table for A and B
We create a truth table to determine the truth values of and for all possible combinations of , , and . We also need the truth value of .
| T | T | T | F | T | T |
| T | T | F | F | T | F |
| T | F | T | F | F | T |
| T | F | F | F | F | F |
| F | T | T | T | F | T |
| F | T | F | T | F | T |
| F | F | T | T | F | T |
| F | F | F | T | F | T |
Step 3: Construct the truth table for and
Now, we find the truth values of the conditional statements and using the truth values of and from the previous step.
| T | T | T | F | T | T | T | T |
| T | T | F | F | T | F | F | T |
| T | F | T | F | F | T | T | F |
| T | F | F | F | F | F | T | T |
| F | T | T | T | F | T | T | F |
| F | T | F | T | F | T | T | F |
| F | F | T | T | F | T | T | F |
| F | F | F | T | F | T | T | F |
Step 4: Construct the truth table for
Finally, we find the truth value of the disjunction using the truth values of and from the previous step.
| T | T | T | F | T | T | T | T | T |
| T | T | F | F | T | F | F | T | T |
| T | F | T | F | F | T | T | F | T |
| T | F | F | F | F | F | T | T | T |
| F | T | T | T | F | T | T | F | T |
| F | T | F | T | F | T | T | F | T |
| F | F | T | T | F | T | T | F | T |
| F | F | F | T | F | T | T | F | T |
Step 5: Analyze the truth table
Observe that the last column, which represents the truth value of the given statement, always has the value 'T' (True) regardless of the values of , , and . Therefore, the given statement is a tautology.
Common Mistakes & Tips
- Be very careful when evaluating conditional statements. Remember that is only false when is true and is false.
- When creating truth tables, ensure that you consider all possible combinations of truth values for the component statements.
- Recognize the logical equivalence: is always a tautology.
Summary
By breaking down the given conditional statement into smaller components and constructing a truth table, we have shown that the statement is always true, regardless of the truth values of , , and . Therefore, the given statement is a tautology.
Final Answer
The final answer is \boxed{a tautology}, which corresponds to option (A).