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

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
Transcript text: Determine all possible value combinations of $A, B$, and $C$ so the formula is true: $(A \wedge B) \vee((C \vee A) \wedge B)$ AND FORM A TABLE
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Solution

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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.

Step 1: List All Possible Combinations of \( A, B, \) and \( C \)

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

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

Step 2: Evaluate the Formula for Each Combination

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

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

Step 3: Identify the Combinations that Make the Formula True

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

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

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