If each of the statements `p to ~q , ~ r to q` and p are true then which of the following is NOT true ?

Options

A. q is false
B. r is true
C. `r to q` is false
D. `r^^~q` is false

Explanation

To solve the problem step-by-step, we will analyze the given statements and their implications.

### Step 1: Identify the Statements
We have three statements:
1. \( p \to \neg q \) (p implies not q)
2. \( \neg r \to q \) (not r implies q)
3. \( p \) (p is true)

We need to determine which of the following statements is NOT true given that all three statements are true.

### Step 2: Construct the Truth Table
We need to create a truth table for the variables \( p \), \( q \), and \( r \). Since there are three variables, there will be \( 2^3 = 8 \) possible combinations of truth values.

| p | q | r | \( \neg q \) | \( \neg r \) | \( p \to \neg q \) | \( \neg r \to q \) |
|-----|-----|-----|---------------|---------------|---------------------|---------------------|
| T | T | T | F | F | F | T |
| T | T | F | F | T | F | T |
| T | F | T | T | F | T | F |
| T | F | F | T | T | T | T |
| F | T | T | F | F | T | T |
| F | T | F | F | T | T | T |
| F | F | T | T | F | T | F |
| F | F | F | T | T | T | T |

### Step 3: Evaluate the Implications
Now we need to evaluate the implications based on the truth values:

1. **For \( p \to \neg q \)**:
- This implication is false only when \( p \) is true and \( \neg q \) is false (i.e., \( q \) is true).
- From the table, this is true in the cases where \( p \) is true and \( q \) is false.

2. **For \( \neg r \to q \)**:
- This implication is false only when \( \neg r \) is true (i.e., \( r \) is false) and \( q \) is false.
- From the table, this is true when \( r \) is true or when both \( \neg r \) is false and \( q \) is true.

### Step 4: Identify Valid Cases
Now we look for cases where all three statements are true:
- From the truth table, we find that the valid cases are:
- Case 3: \( p = T, q = F, r = T \)
- Case 4: \( p = T, q = F, r = F \)

### Step 5: Check the Options
Now we check the options provided in the original question to find which is NOT true:

1. **Option 1: \( q \) is false** - This is true in both valid cases.
2. **Option 2: \( r \) is true** - This is NOT true in Case 4 where \( r \) is false.
3. **Option 3: (other options not provided)** - We cannot evaluate without additional options.

### Conclusion
The statement that is NOT true given the conditions is:
- **Option 2: \( r \) is true**.