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Multiplexers and Demultiplexers are devices that move signals from wire to wire, based on a given control signal.
Switching Circuits 
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.
Input/Output Relationships 
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.