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Multiplexer is a device that allows multiple input lines to connect to a single output. There is a control mechanism that can select active line between these inputs.
In this schematic, the control logic is on the left - the 3 on/off buttons with NOT gates directly below each one - allowing selection of up to 8 lines (3-bit binary, 0-7). Bottom left corner will output the selected line on a digital display, just for convenience - this is not part of the multiplexer logic. The series of AND gates, however (2 columns of 8 gates each) are part of the multiplexer logic - they create the possible combinations of the binary numbers 0-7. Note, other gates may be used, but I chose AND gates as it made sense to me to represent each number as a 3-input AND gate (hence 2 columns of ANDs - 1st and 2nd stage, turning it into a 3-input AND gate) between either the inputs, the inverse of the inputs (hence the NOT gates below each input), or some combination of the two.
The output from the 2nd column of AND gates is then ANDed with each input line being controlled by the multiplexer (right-most column of AND gates), with an LED next to the on/off button for each line to indicate when this line is selected (again, this is mostly for convenience, and is not part of the logic). Each line can then be individually turned on/off, and multiple lines can be on at the same time, but only the selected line will go through. The output is indicated by a lamp in the lower right-hand corner, and is connected to the ANDed output of the input lines and control lines via a multi-input OR gate (represented by a column of OR gates on the right), allowing any of the selected lines to go through.
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