A half adder adds two 1-bit numbers. It has two outputs: the sum and the carry, which goes to the next column. A half adder cannot be used for numbers of more than one bit. When adding numbers of multiple columns there may be a carry in from the previous column, and the half adder doesn't handle this. A full adder adds two 1-bit numbers, with the additional input of a carry in. It also has two outputs, the sum and the carry out.
Multiple full adders can be used to add multiple bit numbers. The carry in to the rightmost column is zero. The carry in to every other column is the carry out of the previous column. This is called a ripple carry adder. It had the advantage of being simple but it has the disadvantage of being slow, since each full adder must wait for the previous one to finish. There are other designs for adders that are faster.
A decoder is a circuit that uses its inputs to select one of its outputs. Only the selected output will be asserted; all other outputs will be zero. A decoder with n inputs can have at most 2n outputs. Thus a decoder with 2 inputs can have at most 4 outputs (called a 2x4 decoder) and a decoder with 3 inputs can have at most 8 outputs (called a 3x8 decoder).
A multiplexer has many input lines and one output line. The signal from one input line will be directed to the output line. The input line is chosen based on the signals which are carried to the multiplexer on another set of input lines called control lines. Multiplexers are sometimes called selectors because they choose or select one of their inputs.
The number of control lines needed depends on the number of input lines. A multiplexer with 2 control lines can select from 4 input lines, a multiplexer with 3 control lines can select from 8 input lines. In general, a multiplexer with n control lines can select from up to 2ninput lines.
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