`p implies q` is false if :

Options

A. p is false, q is true
B. p is false, q is false
C. p is true, q is false
D. p is true, q is true

Answer

To determine when the statement "P implies Q" is false, we can analyze the logical implication using a truth table. Here’s a step-by-step solution:

### Step 1: Understand the Implication
The statement "P implies Q" (denoted as P → Q) means that if P is true, then Q must also be true. The implication is only false when P is true and Q is false.

### Step 2: Create a Truth Table
We will create a truth table with three columns: P, Q, and P implies Q (P → Q). We will have four rows to cover all combinations of truth values for P and Q.

| P | Q | P → Q |
|-------|-------|-------|
| True | True | True |
| True | False | False |
| False | True | True |
| False | False | True |

### Step 3: Fill in the Truth Table
1. **First Row (P = True, Q = True)**:
- Since both P and Q are true, P → Q is true.
2. **Second Row (P = True, Q = False)**:
- Here, P is true but Q is false, so P → Q is false.
3. **Third Row (P = False, Q = True)**:
- P is false, regardless of Q, P → Q is true.
4. **Fourth Row (P = False, Q = False)**:
- Again, P is false, so P → Q is true.

### Step 4: Identify When P → Q is False
From the truth table, we see that the only case where P → Q is false is when P is true and Q is false.

### Conclusion
Thus, "P implies Q" is false if P is true and Q is false.