To create a truth table for combined statements in logic, we need to follow some steps and know the basic symbols used. A combined statement is made by putting simple statements together using logic operators.

Basic Symbols in Logic

1. Simple Statements: These are shown with letters like $P$, $Q$, and $R$.

2. Logic Operators:

   • Negation ($\neg$): Means "not".
   • Conjunction ($\land$): Means "and".
   • Disjunction ($\lor$): Means "or".
   • Implication ($\rightarrow$): Means "if... then".
   • Biconditional ($\leftrightarrow$): Means "if and only if".

Steps to Make a Truth Table

1. Find the Simple Statements: Figure out the basic propositions in the combined statement.

2. List Possible Truth Value Combinations: If there are $n$ simple statements, you will have $2^n$ combinations of truth values. For example:

   • 2 statements mean $2^2 = 4$ combinations: (True, True), (True, False), (False, True), (False, False)
   • 3 statements mean $2^3 = 8$ combinations:
     • (True, True, True)
     • (True, True, False)
     • (True, False, True)
     • (True, False, False)
     • (False, True, True)
     • (False, True, False)
     • (False, False, True)
     • (False, False, False)

3. Make the Header: The first row of the truth table will show headers for each simple statement and the combined statement.

4. Fill in the Truth Values: For each combination of truth values, calculate the truth value of the combined statement using the logic operators.

Example: Making a Truth Table

Let’s look at the combined statement $P \land (Q \lor R)$. Here’s how we do it:

1. Find Simple Statements: $P$, $Q$, $R$.

2. List Combinations: For 3 variables, we have 8 combinations:

   • (True, True, True)
   • (True, True, False)
   • (True, False, True)
   • (True, False, False)
   • (False, True, True)
   • (False, True, False)
   • (False, False, True)
   • (False, False, False)

3. Make Header:
   | $P$ | $Q$ | $R$ | $Q \lor R$ | $P \land (Q \lor R)$ | |-----|-----|-----|-------------|-----------------------|

4. Complete the Table:

   • Fill in the columns for $Q \lor R$ and $P \land (Q \lor R)$ based on the truth values.

   | $P$ | $Q$ | $R$ | $Q \lor R$ | $P \land (Q \lor R)$ | |-----|-----|-----|-------------|-----------------------| | T | T | T | T | T | | T | T | F | T | T | | T | F | T | T | T | | T | F | F | F | F | | F | T | T | T | F | | F | T | F | T | F | | F | F | T | T | F | | F | F | F | F | F |

Conclusion

Truth tables are useful tools in logic. They help show the clear relationships and truth values of statements. By breaking down complex logic expressions into simpler parts, we can understand them better.