Question

Is the statement \(p \Rightarrow q\) equivalent to the statement \(\neg p \Rightarrow \neg q\) ?

Short answer

No, the statement \(p \Rightarrow q\) is not equivalent to \(\neg p \Rightarrow \neg q\).

Step-by-step solution

Understanding Implication

The statement \( p \Rightarrow q \) is an implication which is true if either the statement \( p \) is false or the statement \( q \) is true. The only case when \( p \Rightarrow q \) is false is when \( p \) is true and \( q \) is false.

Understanding Contrapositive

The statement \( eg p \Rightarrow eg q \) is the contrapositive of \( p \Rightarrow q \). This means it takes the opposite truth values, meaning it is only false when \( eg p \) is true and \( eg q \) is false, corresponding to when \( p \) is false and \( q \) is true. This, however, is a misunderstanding. It should correspond exactly to \( p \Rightarrow q \).

Determine Logical Equivalence

To determine if \( p \Rightarrow q \) and \( eg p \Rightarrow eg q \) are equivalent, consider when they are both true or false. As \( eg p \Rightarrow eg q \) contradicts the truth pattern that \( p \Rightarrow q \) establishes (i.e., it is not a logical equivalence), let's resolve this using formal logic rules.

Use Logical Analysis

To conclusively determine, we show if both are logically equivalent: - Apply \( p = 0 \) or \( q = 1 \), then both expressions evaluate true. Conversely at \( p = 1, q = 0 \), expression 1 is false and expression 2 is true.

Conclusion

Since their truth values do not align in all scenarios, \( p \Rightarrow q \) is not logically equivalent to \( eg p \Rightarrow eg q \). The truth condition when both are legitimate is for a contrapositive, yet this wasn't contrapositive but negation.

Key concepts

Implication Statement

An implication statement, often denoted as \( p \Rightarrow q \), is a logical statement that asserts "if \( p \), then \( q \)". Here, \( p \) and \( q \) are individual propositions that can be either true or false. The truth of the implication is crucial and follows a specific pattern: the statement is false only when \( p \) is true and \( q \) is false. In any other case, whether \( q \) is true or \( p \) is false, the implication holds as true.
Consider these simple bullet points for an implication statement:

• True when both \( p \) is false or \( q \) is true.
• True if both \( p \) and \( q \) are true.
• False only when \( p \) is true and \( q \) is false.

This understanding helps to clearly interpret logical statements when trying to deduce logical equivalence or perform truth table analysis.

Contrapositive

The contrapositive of a given implication \( p \Rightarrow q \) flips and negates both parts of the statement. It is written as \( eg q \Rightarrow eg p \). The beauty of contrapositives is that they are logically equivalent to the original implication. This means whenever \( p \Rightarrow q \) is true, so is \( eg q \Rightarrow eg p \), and vice versa.
Let's break down why the contrapositive is so powerful:

• It affirms the same truth value as the original statement.
• Whenever the implication holds under any particular situation, its contrapositive holds too.
• Helps in proving logically complex structures by flipping terms.

Understanding contrapositives allows mathematicians and logicians to prove equivalence and other logical properties more efficiently.

Truth Table Analysis

Performing a truth table analysis is an effective way to visualize and determine logical equivalence between statements like implications and their contrapositives. It involves laying out all possible truth values for your propositions \( p \) and \( q \) and calculating the results for each scenario.
Here's how truth table analysis is approached:

• List all possible combinations of truth values for \( p \) and \( q \).
• Evaluate the truth value of \( p \Rightarrow q \) for each combination.
• Compare with the truth value of \( eg p \Rightarrow eg q \), the supposed contrapositive.

Through this approach, the [logical equivalence](https://en.wikipedia.org/wiki/Logical_equivalence) between different statements like \( p \Rightarrow q \) and its contrapositive \( eg q \Rightarrow eg p \) can be accurately verified. Truth table analysis ensures a clear and precise understanding of logical relationships.