Introduction:
The difference between a half adder and a full adder lies in the number of inputs they have. Both half adder and full adder are combinational logic circuits used for arithmetic operations. They are used to add binary numbers and produce sum and carry outputs.

Half Adder:
A half adder is a basic digital circuit that adds two single-bit binary numbers A and B. It has two inputs, A and B, and two outputs, Sum (S) and Carry (C). The sum output represents the least significant bit of the addition, while the carry output represents the carry generated from adding both bits.

Full Adder:
A full adder is an extension of a half adder that can add three input bits - A, B, and an incoming carry (Cin). It has three inputs, A, B, and Cin, and two outputs, Sum (S), and Carry (Cout). The sum output represents the least significant bit of the addition, while the carry output represents the carry generated from adding all three bits.

Differences:
The key difference between a half adder and a full adder can be summarized as follows:

1. Number of inputs:
- A half adder has two inputs, A and B, representing the bits to be added.
- A full adder has three inputs, A, B, and Cin, where Cin is the incoming carry from the previous stage.

2. Number of outputs:
- A half adder has two outputs, Sum (S) and Carry (C).
- A full adder also has two outputs, Sum (S), and Carry (Cout).

3. Carry propagation:
- In a half adder, there is no provision to consider any incoming carry.
- In a full adder, the third input (Cin) allows for the consideration of an incoming carry.

4. Cascading:
- A half adder cannot be directly cascaded to perform multi-bit addition since it does not consider the carry generated from the previous stage.
- A full adder can be cascaded to perform multi-bit addition by using the carry output (Cout) as the Cin input for the next stage.

Conclusion:
In summary, a half adder is a basic circuit that adds two bits without considering any incoming carry, while a full adder is an extended circuit that adds three bits, considering both the bits and the incoming carry. The additional input in a full adder allows for cascading multiple full adders to perform addition of multi-bit numbers.