To establish the truth that a conditional statement is not logically equivalent to its inverse using truth tables, we need to:

1. Define the conditional statement (if $p$, then $q$) as $p \rightarrow q$.
2. Define the inverse of the conditional statement (if not $p$, then not $q$) as $\neg p \rightarrow \neg q$.

Step-by-Step Process:

1. Define the symbols:

• $p$: Proposition 1.
• $q$: Proposition 2.
• $p \rightarrow q$: Conditional statement.
• $\neg p \rightarrow \neg q$: Inverse of the conditional statement.

2. Construct the truth table:

We’ll build a truth table to evaluate the truth values of both $p \rightarrow q$ and $\neg p \rightarrow \neg q$ and compare them.

$p$	$q$	$\neg p$	$\neg q$	$p \rightarrow q$	$\neg p \rightarrow \neg q$
T	T	F	F	T	T
T	F	F	T	F	T
F	T	T	F	T	F
F	F	T	T	T	T

3. Analyze the truth table:

• $p \rightarrow q$ (the conditional statement) and $\neg p \rightarrow \neg q$ (the inverse) are not logically equivalent, because their truth values differ in the third row.
• In row 3, $p \rightarrow q$ is true, while $\neg p \rightarrow \neg q$ is false.

This difference confirms that a conditional statement is not logically equivalent to its inverse.

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Would you like further clarification or details on any step? Here are 5 related questions to deepen your understanding:

1. How does the converse of a conditional statement differ from the inverse?
2. Can you find an example of a conditional statement and its inverse in real-life situations?
3. How is the contrapositive of a conditional statement related to its truth value?
4. Why is it important to understand the difference between logical equivalence and non-equivalence?
5. How would the truth table change for the biconditional statement $p \leftrightarrow q$?

Tip: The contrapositive of a conditional statement is always logically equivalent to the original conditional.