The image shows a partially completed truth table for the logical statement $\sim b \leftrightarrow (b \rightarrow a)$. Let's break this down step by step and complete the table:

Symbols:

• $a$: A proposition (True or False)
• $b$: Another proposition (True or False)
• $\sim b$: Negation of $b$
• $b \rightarrow a$: Conditional statement ("if $b$, then $a$")
• $\sim b \leftrightarrow (b \rightarrow a)$: Biconditional ("$\sim b$ if and only if $b \rightarrow a$")

Truth Table Columns:

1. $a$
2. $b$
3. $b \rightarrow a$ (Conditional statement)
4. $\sim b$ (Negation of $b$)
5. $\sim b \leftrightarrow (b \rightarrow a)$ (Biconditional)

Truth Table Logic:

Explanation:

• Column 3 $b \rightarrow a$: This is true if $b$ is false or if both $b$ and $a$ are true.
• Column 4 $\sim b$: The negation of $b$.
• Column 5 $\sim b \leftrightarrow (b \rightarrow a)$: True if both sides are either true or false (biconditional).

With this, the table is now complete. Would you like more details on any of the steps or logic?

Here are 5 questions to explore the topic further:

1. How does the biconditional differ from an implication?
2. Can you explain the difference between $b \rightarrow a$ and $a \rightarrow b$?
3. How does the truth value of $b \rightarrow a$ change depending on $a$ and $b$?
4. Why is $\sim b \leftrightarrow (b \rightarrow a)$ false when both $a$ and $b$ are true?
5. How would the truth table change if the statement was $a \leftrightarrow (b \rightarrow a)$?

Tip:

When working with truth tables, it's useful to remember that $p \rightarrow q$ is only false when $p$ is true and $q$ is false, and biconditionals are only true when both sides have the same truth value.