In Boolean algebra A.B =Y implies that :

Options

A. product of A and B is Y
B. Y exists when A exists or B exists
C. Y exists when both A and B exist but not when only A or B exists
D. Y exists when A or B exists but not both A and B exist.

Answer

To solve the question "In Boolean algebra A.B = Y implies that:", we need to analyze the implications of the expression A.B = Y in the context of Boolean algebra.

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression A.B = Y means that the output Y is the result of the logical AND operation between A and B. In Boolean algebra, the AND operation is represented by the dot (.) operator.

**Hint**: Recall that in Boolean algebra, the AND operation results in true (1) only when both inputs are true (1).

2. **Evaluating the Truth Table**:
Let's create a truth table for A and B to understand the output Y:

| A | B | A.B (Y) |
|---|---|---------|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |

From the truth table, we can see that Y (A.B) is true (1) only when both A and B are true (1).

**Hint**: Remember that the output Y is only true when both inputs A and B are true.

3. **Analyzing the Statements**:
Now, let's analyze the provided statements based on our understanding of the AND operation:

- **Statement 1**: "The product of A and B is Y."
- This statement is misleading because it suggests that Y is always equal to the product of A and B, which is not true unless both A and B are 1.

- **Statement 2**: "Y exists when A exists and B exists."
- This statement is incorrect because Y can be 0 even if A and B exist (i.e., they can both be 0 or one can be 0).

- **Statement 3**: "Y exists when both A and B exist."
- This statement is correct. Y (A.B) is true only when both A and B are true.

- **Statement 4**: "Y exists when A and B exist, but not when both A and B exist."
- This statement is incorrect as it contradicts the definition of the AND operation.

**Hint**: Carefully evaluate each statement based on the truth table and the definition of the AND operation.

4. **Conclusion**:
The correct implication of A.B = Y is that Y exists only when both A and B are true. Therefore, the correct answer is Statement 3.

### Final Answer:
The correct implication of A.B = Y is that Y exists when both A and B exist.