Problem 40
Question
Statement- \(1: \sim(p \leftrightarrow \sim q)\) is equivalent to \(p \leftrightarrow q\). Statement- \(2: \sim(p \leftrightarrow \sim q)\) is a tantology (a) Statement- 1 is true, Statement- 2 is true; Statement- 2 is not a correct explanation for Statement-1. (b) Statement- 1 is true, Statement- 2 is false. (c) Statement- 1 is false, Statement- 2 is true. (d) Statement- 1 is true, Statement- 2 is true, Statement- 2 is a correct explanation for statement \(-1\)
Step-by-Step Solution
Verified Answer
(b) Statement- 1 is true, Statement- 2 is false.
1Step 1: Analyze Statement 1
Statement 1 claims that \( \sim(p \leftrightarrow \sim q) \) is equivalent to \( p \leftrightarrow q \). To verify this, calculate the negation of \(p \leftrightarrow \sim q\) which is \((p \land q) \lor (\sim p \land \sim q)\) using the properties of logical equivalences. Compare it to \(p \leftrightarrow q\) which is \((p \land q) \lor (\sim p \land \sim q)\). Since these expressions are equivalent, Statement 1 is true.
2Step 2: Analyze Statement 2
A tautology is a formula that is always true regardless of the truth values of its components. Re-examine \( \sim(p \leftrightarrow \sim q) \). Since it resulted in \((p \land q) \lor (\sim p \land \sim q)\), evaluate whether this formula is a tautology. Check truth tables: \((p \land q) \lor (\sim p \land \sim q)\) is true if both \(p\) and \(q\) have the same truth value. Therefore, it is not always true but matches the definition of a biconditional. Thus, Statement 2 is false.
3Step 3: Determine Correct Option
Based on the analysis: Statement 1 is true and Statement 2 is false, which corresponds to option (b). Therefore, the correct answer is (b).
Key Concepts
Understanding TautologyDecoding Truth TablesThe Role of the Biconditional in Logic
Understanding Tautology
A tautology is a special kind of logical statement that is always true, no matter the truth values of its components. It means that regardless of when or how you evaluate the expression, its outcome is consistently true.
To determine if a logical statement is a tautology, we use logical equivalences and sometimes truth tables to see if all possible scenarios lead to a true outcome.
Think of a tautology as being universally applicable in logic because there are no circumstances under which it can evaluate to false. In the context of complex statements, a tautology is invaluable as it reassures consistency and unity in logical reasoning.
To determine if a logical statement is a tautology, we use logical equivalences and sometimes truth tables to see if all possible scenarios lead to a true outcome.
Think of a tautology as being universally applicable in logic because there are no circumstances under which it can evaluate to false. In the context of complex statements, a tautology is invaluable as it reassures consistency and unity in logical reasoning.
Decoding Truth Tables
Truth tables are systematic tools used to evaluate the truth values of logical expressions. They list all possible combinations of truth values for given propositions and determine the overall truth of the compound statement.
To construct a truth table:
By examining these tables, you can decide if a statement is a tautology, a contradiction, or simply a conditional true statement.
To construct a truth table:
- Identify all unique propositions within the expression.
- Create rows for every possible combination of truth values (true or false) for these propositions.
- Compute the expression for each combination to fill out the table.
By examining these tables, you can decide if a statement is a tautology, a contradiction, or simply a conditional true statement.
The Role of the Biconditional in Logic
The biconditional, represented by the symbol \( \leftrightarrow \), plays a crucial role in logical expressions. This operator represents 'if and only if,' meaning both parts of the compound statement have the same truth value for the entire statement to be true.
A biconditional statement \( p \leftrightarrow q \) only evaluates to true when both \( p \) and \( q \) are either both true or both false. In other words, it establishes a perfect balance between the two propositions, reflecting a mutual agreement.
The essence of the biconditional lies in its demand for equality in truth values, making it essential to verify logical equivalences and relationships in compound expressions. It ensures logical consistency and is used extensively in mathematical proofs and formal logic.
A biconditional statement \( p \leftrightarrow q \) only evaluates to true when both \( p \) and \( q \) are either both true or both false. In other words, it establishes a perfect balance between the two propositions, reflecting a mutual agreement.
The essence of the biconditional lies in its demand for equality in truth values, making it essential to verify logical equivalences and relationships in compound expressions. It ensures logical consistency and is used extensively in mathematical proofs and formal logic.
Other exercises in this chapter
Problem 38
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