Question

\((p \Rightarrow q) \Leftrightarrow(\sim q \Rightarrow \sim p)\) is a (a) contradiction (b) tautology (c) both tautology \& contradiction (d) None of above

Short answer

The given statement \((p \Rightarrow q) \Leftrightarrow (\sim q \Rightarrow \sim p)\) is a (b) tautology, as it is true in all possible cases.

Step-by-step solution

Create truth table for p and q values

First, let's create a truth table for all possible combinations of p and q: | p | q | |---|---| | T | T | | T | F | | F | T | | F | F |

Determine the values for the given statement

Next, calculate the values of the implications \((p \Rightarrow q)\) and \((\sim q \Rightarrow \sim p)\) for each combination of p and q: | p | q | p => q | ~q | ~p | ~q => ~p | |---|---|--------|----|----|----------| | T | T | T | F | F | T | | T | F | F | T | F | F | | F | T | T | F | T | T | | F | F | T | T | T | T | Note that \((\sim q \Rightarrow \sim p)\) is the contrapositive of \((p \Rightarrow q)\).

Check if the statement is a tautology, contradiction, both, or neither

To check if the statement is a tautology (always true), contradiction (always false), both, or neither, compare the values of the implications \((p \Rightarrow q)\) and \((\sim q \Rightarrow \sim p)\): | p | q | p => q | ~q => ~p | (p => q) <=> (~q => ~p) | |---|---|--------|----------|-------------------------| | T | T | T | T | T | | T | F | F | F | T | | F | T | T | T | T | | F | F | T | T | T | Since \((p \Rightarrow q) \Leftrightarrow (\sim q \Rightarrow \sim p)\) is true in all possible cases, it is a: (a) contradiction: no, it's not always false (b) tautology: yes, it's always true (c) both tautology & contradiction: no, it can't be both at the same time (d) None of above: no, it fits the definition of a tautology So, the correct answer is (b) tautology.

Key concepts

Implication

Implication is a fundamental concept in logic, represented by the symbol \( \Rightarrow \). It expresses a conditional statement or an "if-then" scenario. Such a statement is of the form "If \( p \), then \( q \)" and written as \( p \Rightarrow q \).
- **In logic,** a statement \( p \Rightarrow q \) is considered false only when \( p \) is true and \( q \) is false. If both \( p \) and \( q \) are true or \( p \) is false for any value of \( q \), the implication holds true.
Understanding implications is crucial for logical reasoning, as it helps in shaping hypotheses and drawing conclusions in arguments.

Truth Table

A truth table is a useful tool in logic that shows how the truth value of a statement depends on the truth values of its components. They are graphical representations that can clarify how logical operations and relationships work.
- **For instance,** a truth table for an implication such as \( p \Rightarrow q \) includes all possible truth value combinations of \( p \) and \( q \). - It also shows the resulting truth value of \( p \Rightarrow q \) based on these combinations.
Using a truth table aids in quick verification of logical statements since each scenario can be easily checked to see if it holds as true or false.

Logic

Logic is the structured study of principles of valid reasoning and argument. It deals with the relationship between concepts and the inference of new information from given facts.
- **In classical logic,** propositions are declared true or false. Logical connectives like "and," "or," "not," and "implies" form compound propositions from simpler ones.
- **Logical reasoning** involves examining these relationships and establishing truths using structured processes. - It's essential in mathematics and computer science, where precise and reliable reasoning is necessary. Learning logic helps individuals think critically and solve complex problems in a variety of fields.

Contrapositive

The contrapositive of an implication \( p \Rightarrow q \) is another statement, \( \sim q \Rightarrow \sim p \). This logically equivalent form flips the roles of \( p \) and \( q \) and negates both.
- **Contrapositives** are important because they always share the same truth value as their original implication. If \( p \Rightarrow q \) is true, then \( \sim q \Rightarrow \sim p \) is also true.
- **Using contrapositive,** one can often simplify or reinterpret logical statements. This concept is particularly handy in mathematical proofs, where demonstrating a contrapositive can sometimes be more straightforward than proving the original statement directly.