Questions: Determine all possible value combinations of A, B, and C so the formula is true: (A AND B) OR ((C OR A) AND B) AND FORM A TABLE

Solution Steps

To determine all possible value combinations of \( A, B \), and \( C \) that make the formula \((A \wedge B) \vee ((C \vee A) \wedge B)\) true, we need to evaluate the formula for all possible combinations of truth values for \( A, B \), and \( C \). We will form a truth table to list all combinations and check which ones satisfy the formula.

We start by listing all possible combinations of the truth values for \( A, B, \) and \( C \). There are \( 2^3 = 8 \) possible combinations:

\[ \begin{array}{ccc} A & B & C \\ \hline \text{True} & \text{True} & \text{True} \\ \text{True} & \text{True} & \text{False} \\ \text{True} & \text{False} & \text{True} \\ \text{True} & \text{False} & \text{False} \\ \text{False} & \text{True} & \text{True} \\ \text{False} & \text{True} & \text{False} \\ \text{False} & \text{False} & \text{True} \\ \text{False} & \text{False} & \text{False} \\ \end{array} \]

Next, we evaluate the formula \((A \wedge B) \vee ((C \vee A) \wedge B)\) for each combination of \( A, B, \) and \( C \):

\[ \begin{array}{cccc} A & B & C & \text{Formula} \\ \hline \text{True} & \text{True} & \text{True} & \text{True} \\ \text{True} & \text{True} & \text{False} & \text{True} \\ \text{True} & \text{False} & \text{True} & \text{False} \\ \text{True} & \text{False} & \text{False} & \text{False} \\ \text{False} & \text{True} & \text{True} & \text{True} \\ \text{False} & \text{True} & \text{False} & \text{False} \\ \text{False} & \text{False} & \text{True} & \text{False} \\ \text{False} & \text{False} & \text{False} & \text{False} \\ \end{array} \]

From the table, we identify the combinations where the formula evaluates to \(\text{True}\):

\[ \begin{array}{ccc} A & B & C \\ \hline \text{True} & \text{True} & \text{True} \\ \text{True} & \text{True} & \text{False} \\ \text{False} & \text{True} & \text{True} \\ \end{array} \]

Final Answer