The compound statement with "And" is false If

Options

A. All its component statements are false
B. Any one of its component are false
C. All its component statements is false
D. Both (1) & (2)

Answer

To determine when a compound statement with "And" is false, we can analyze the conditions under which this occurs. Let's break this down step by step.

### Step-by-Step Solution:

1. **Understanding the "And" Operator**:
The "And" operator (denoted as ∧ in logic) combines two statements, A and B. The compound statement A ∧ B is true only if both A and B are true.

2. **Truth Table Construction**:
We can construct a truth table to visualize the outcomes of the compound statement A ∧ B based on the truth values of A and B.

| A | B | A ∧ B |
|-------|-------|-------|
| True | True | True |
| True | False | False |
| False | True | False |
| False | False | False |

3. **Analyzing the Truth Table**:
From the truth table, we observe the following:
- A ∧ B is true only when both A and B are true.
- A ∧ B is false in the following cases:
- When A is true and B is false (True ∧ False = False)
- When A is false and B is true (False ∧ True = False)
- When both A and B are false (False ∧ False = False)

4. **Conclusion**:
Therefore, we can conclude that the compound statement A ∧ B is false if:
- At least one of the component statements (A or B) is false.
- Specifically, it is false if:
- Both A and B are false, or
- Either A is false or B is false.

### Final Answer:
The compound statement with "And" is false if at least one of its component statements is false.

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