The statement `p Rightarrow p vv q`

Options

A. a tautology
B. a contradiction
C. both a tautology and contradiction
D. None of these

Answer

To determine whether the statement \( p \Rightarrow (p \lor q) \) is a tautology, contradiction, or neither, we will construct a truth table. Here are the steps to solve the problem:

### Step 1: Define the Variables
We have two variables, \( p \) and \( q \). Each variable can either be true (T) or false (F).

### Step 2: Create the Truth Table
We will create a truth table that includes all possible combinations of truth values for \( p \) and \( q \).

| \( p \) | \( q \) | \( p \lor q \) | \( p \Rightarrow (p \lor q) \) |
|---------|---------|----------------|-------------------------------|
| T | T | T | T |
| T | F | T | T |
| F | T | T | T |
| F | F | F | T |

### Step 3: Calculate \( p \lor q \)
- For \( p = T \) and \( q = T \): \( p \lor q = T \)
- For \( p = T \) and \( q = F \): \( p \lor q = T \)
- For \( p = F \) and \( q = T \): \( p \lor q = T \)
- For \( p = F \) and \( q = F \): \( p \lor q = F \)

### Step 4: Calculate \( p \Rightarrow (p \lor q) \)
The implication \( p \Rightarrow (p \lor q) \) is defined as false only when \( p \) is true and \( (p \lor q) \) is false. Otherwise, it is true.

- For \( p = T \) and \( q = T \): \( p \Rightarrow (p \lor q) = T \)
- For \( p = T \) and \( q = F \): \( p \Rightarrow (p \lor q) = T \)
- For \( p = F \) and \( q = T \): \( p \Rightarrow (p \lor q) = T \)
- For \( p = F \) and \( q = F \): \( p \Rightarrow (p \lor q) = T \)

### Step 5: Analyze the Results
From the truth table, we see that \( p \Rightarrow (p \lor q) \) is true for all combinations of truth values for \( p \) and \( q \). Therefore, the statement is a tautology.

### Conclusion
The statement \( p \Rightarrow (p \lor q) \) is a tautology.

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