To determine the operations represented by \( \wedge \) (cross) and \( o+ \) (plus) in the expression \( (p \wedge q) \, o+ \, (p \, o+ \, q) \) such that it is a tautology, we will analyze the truth values using a truth table.
### Step 1: Define the operations
We start by defining the operations:
- \( p \wedge q \): This represents the logical AND operation.
- \( p \, o+ \, q \): This represents the logical OR operation.
### Step 2: Create a truth table
We will create a truth table for \( p \) and \( q \) to evaluate \( (p \wedge q) \, o+ \, (p \, o+ \, q) \).
| \( p \) | \( q \) | \( p \wedge q \) | \( p \, o+ \, q \) | \( (p \wedge q) \, o+ \, (p \, o+ \, q) \) |
|---------|---------|-------------------|---------------------|-------------------------------------------|
| T | T | T | T | T |
| T | F | F | T | T |
| F | T | F | T | T |
| F | F | F | F | F |
### Step 3: Analyze the truth table
From the truth table:
- The expression \( (p \wedge q) \, o+ \, (p \, o+ \, q) \) is true for all combinations of \( p \) and \( q \) except for \( (F, F) \).
- For the expression to be a tautology, it must be true for all possible truth values of \( p \) and \( q \).
### Step 4: Determine the operations
To satisfy the condition of being a tautology, we need to analyze the possible operations for \( o+ \):
1. If \( o+ \) is OR (denoted as \( \vee \)), then the expression becomes:
\[
(p \wedge q) \vee (p \vee q)
\]
This will yield true in all cases except when both \( p \) and \( q \) are false (which is not a tautology).
2. If \( o+ \) is implication (denoted as \( \Rightarrow \)), then the expression becomes:
\[
(p \wedge q) \Rightarrow (p \vee q)
\]
This is true in all cases because if \( p \wedge q \) is true, then \( p \vee q \) must also be true.
### Conclusion
Thus, for the expression \( (p \wedge q) \, o+ \, (p \, o+ \, q) \) to be a tautology, the operations must be:
- \( \wedge \) is AND
- \( o+ \) is implication
### Final Answer
If \( (p \wedge q) \, o+ \, (p \, o+ \, q) \) is a tautology, then \( o+ \) is implication.
