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Multiplexers and Demultiplexers are devices that move signals from wire to wire, based on a given control signal.
Let's say that we have 2 inputs to a given circuit, and we only have one output. We would like the two inputs to "share" the output, and we therefore need to switch from one input to the other. We have inputs I1 and I2, output O, and a control wire, C. We can say that if C=1, then we connect I1 to O. However, if C=0, then we connect I2 to O. This is a multiplexer.
Generally, multiplexers are circuits behaving like a controlled rotary switch, i.e. any one of a number of inputs may be selected as output. In digital electronics, a multiplexer is a combination of logic gates resulting into circuits with two or more inputs (data inputs) and one output. The selection of the channel to be read into the output is controlled by supplying a specific digital word to a different set of inputs (select inputs). A typical 4 input channels (D3-D0) digital multiplexer, and its corresponding truth table.
The active input channel is selected by supplying the appropriate code to select inputs (C1, C0).
Similar to the situation above, let's say we have a circuit with a single input (I), two outputs (O1 and O2), and a control wire, C. When C=1, we connect I to O1, and when C=0, we connect I to O2. This is a demultiplexer.