If p is true, q is false and r is false, then which of the following is true :

Options

A. `(p vv q) implies r`
B. `p ^^ ~| (q vv r)`
C. `(p vv q) ^^ r`
D. `p implies ~| (q implies r)`

Answer

To determine which of the given options is true based on the truth values of \( p \), \( q \), and \( r \), we will evaluate each option step by step.

### Given:
- \( p \) is true (T)
- \( q \) is false (F)
- \( r \) is false (F)

### Options to Evaluate:
1. \( p \lor q \implies r \)
2. \( p \land \neg q \lor r \)
3. \( p \lor q \land r \)
4. \( p \implies \neg q \implies r \)

### Step 1: Evaluate Option A: \( p \lor q \implies r \)

- \( p \lor q \) evaluates to \( T \lor F = T \).
- Now evaluate \( T \implies F \).
- The implication \( T \implies F \) is false.

**Result for Option A: False**

### Step 2: Evaluate Option B: \( p \land \neg q \lor r \)

- First, evaluate \( \neg q \): since \( q \) is false, \( \neg q \) is true (T).
- Now evaluate \( p \land \neg q \): \( T \land T = T \).
- Now evaluate \( T \lor r \): \( T \lor F = T \).

**Result for Option B: True**

### Step 3: Evaluate Option C: \( p \lor q \land r \)

- First, evaluate \( q \land r \): \( F \land F = F \).
- Now evaluate \( p \lor (q \land r) \): \( T \lor F = T \).

**Result for Option C: True**

### Step 4: Evaluate Option D: \( p \implies \neg q \implies r \)

- First, evaluate \( \neg q \): since \( q \) is false, \( \neg q \) is true (T).
- Now evaluate \( p \implies \neg q \): \( T \implies T = T \).
- Now evaluate \( T \implies r \): \( T \implies F \) is false.

**Result for Option D: False**

### Summary of Results:
- Option A: False
- Option B: True
- Option C: True
- Option D: False

### Conclusion:
The options that are true are B and C. However, since the question asks for which of the following is true and we have multiple true options, we will select the first true option encountered.

**Final Answer: Option B is true.**

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