Read this chapter, which describes the design of several components using logic gates, including adders, encoders and decoders, multiplexers, and demultiplexers. This chapter also mentions ladder logic. If you are not familiar with ladder logic, you may also optionally read Chapter 6.1 and 6.2 as a reference. Note that ladder logic is not generally used in computer design and may be omitted. For the one input decoder, note that for a 0 input, the D0 output is a 1 and when the input is a 1, the D1 output is a 1. All the other decoders work the same way. One output line is a 1and the rest are 0's, indicating which binary number has been placed on the input lines. So an input in binary of the number 6 would cause D6 to be a 1.
A multiplexer, abbreviated mux, is a device that has multiple inputs and one output. The schematic symbol for multiplexers is
The truth table for a 2-to-1 multiplexer is
Using a 1-to-2 decoder as part of the circuit, we can express this circuit easily.
Multiplexers can also be expanded with the same naming conventions as demultiplexers. A 4-to-1 multiplexer circuit is
That is the formal definition of a multiplexer. Informally, there is a lot of confusion. Both demultiplexers and multiplexers have similar names, abbreviations, schematic symbols and circuits, so confusion is easy. The term multiplexer, and the abbreviation mux, are often used to also mean a demultiplexer, or a multiplexer and a demultiplexer working together. So when you hear about a multiplexer, it may mean something quite different.