Half Adder:
A half adder is a logical circuit that adds two single-bit binary numbers A and B. It has two inputs, A and B, and two outputs, the sum (S) and the carry (C).

Inputs:
In a half adder, the inputs A and B represent the two bits that need to be added.

Carry:
The carry output of a half adder is a logic gate that determines if there is a carry generated when adding A and B together. It indicates whether there is a need to carry over to the next bit when adding multiple bits.

Logic Gates:
Logic gates are the fundamental building blocks of digital circuits. They perform logical operations on one or more binary inputs to produce a single binary output.

AND Gate:
An AND gate is a logic gate that outputs a 1 if and only if all of its inputs are 1. Otherwise, it outputs a 0. The truth table for an AND gate is as follows:

```
A | B | Output
--------------
0 | 0 | 0
0 | 1 | 0
1 | 0 | 0
1 | 1 | 1
```

Explanation:
In a half adder, the carry output is determined by the AND operation between the inputs A and B. The carry output is 1 only when both inputs A and B are 1, indicating that a carry is generated.

Let's consider the inputs A and B:

```
A | B
-----
0 | 0
0 | 1
1 | 0
1 | 1
```

When A and B are both 0, the carry output is 0. When A is 0 and B is 1, or A is 1 and B is 0, the carry output is also 0. However, when A and B are both 1, the carry output is 1.

Therefore, the carry output of a half adder is given by A AND B, which corresponds to option 'A'.