In the Boolean algebra, which gate is expressed as `Y = overline(A + B)`.

Options

A. `OR`
B. `NAND`
C. `AND`
D. `NOR`

Explanation

To solve the question regarding which gate is expressed as `Y = overline(A + B)`, we can follow these steps:

### Step 1: Understand the expression
The expression `Y = overline(A + B)` indicates that we are dealing with a logical operation involving two variables, A and B. The `+` symbol in Boolean algebra typically represents the OR operation.

### Step 2: Apply De Morgan's Theorem
According to De Morgan's Theorem, the negation of a disjunction (OR operation) can be expressed as the conjunction (AND operation) of the negations. Thus, we can rewrite the expression:
\[ Y = \overline{A + B} = \overline{A} \cdot \overline{B} \]
This means that Y is true only when both A and B are false.

### Step 3: Identify the corresponding gate
The expression `Y = overline(A + B)` corresponds to the NOR gate. The NOR gate outputs true (1) only when both inputs are false (0).

### Step 4: Verify with a truth table
To confirm, we can create a truth table for the expression `Y = overline(A + B)`:

| A | B | A + B | Y = overline(A + B) |
|---|---|-------|---------------------|
| 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 |

From the truth table, we see that Y is 1 only when both A and B are 0, which is characteristic of the NOR gate.

### Conclusion
Thus, the gate expressed as `Y = overline(A + B)` is the **NOR gate**.

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