"If p then q" (where p and q are statements) says

Options

A. If p ts true, then q must be true
B. If p is false, then q must be false
C. Not happening of p has no effect of happening of q
D. Both (1) & (3)

Answer

To solve the statement "If p then q" (denoted as \( p \implies q \)), we can analyze it using a truth table. Here's a step-by-step breakdown of how to create the truth table and interpret the results.

### Step 1: Identify the Variables
We have two statements:
- \( p \): the first statement
- \( q \): the second statement

### Step 2: Determine the Number of Combinations
Since there are two statements, we can calculate the number of possible combinations of truth values (True or False) for these statements. The formula for the number of combinations is \( 2^n \), where \( n \) is the number of variables. Here, \( n = 2 \):
\[ 2^2 = 4 \]
Thus, there are 4 combinations of truth values for \( p \) and \( q \).

### Step 3: Create the Truth Table
Now, we will create a truth table with all possible combinations of truth values for \( p \) and \( q \), along with the output for \( p \implies q \).

| \( p \) | \( q \) | \( p \implies q \) |
|---------|---------|---------------------|
| True | True | True |
| True | False | False |
| False | True | True |
| False | False | True |

### Step 4: Analyze the Output
The output \( p \implies q \) can be interpreted as follows:
- If \( p \) is True and \( q \) is True, then \( p \implies q \) is True.
- If \( p \) is True and \( q \) is False, then \( p \implies q \) is False.
- If \( p \) is False (regardless of whether \( q \) is True or False), then \( p \implies q \) is True.

### Conclusion
From the truth table, we can conclude that:
- The implication \( p \implies q \) is only false when \( p \) is true and \( q \) is false.
- In all other cases, the implication is true.

### Final Answer
The correct interpretation of "If p then q" is that it is true unless \( p \) is true and \( q \) is false.

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